Random nucleotide mutation for nucleotide template counting and assembly

ABSTRACT

A method for determining the number of nucleic acid molecules (NAMs) in a group of NAMs, comprising
         i) obtaining an amplified and mutagenized group of NAMs that was produced by
           a. subjecting the group of NAMs to a chemical mutagenesis which mutates only select nucleic acid bases in the group of NAMs at a rate of 10% to 90% thus forming a group of mutagenized NAMs (mNAMs), and   b. amplifying the group of mNAMs;   
           ii) obtaining sequences of the mNAMs in the group of amplified mNAMs; and   iii) counting the number of different sequences obtained in step (ii) to determine the number of unique mNAMs in the group of mNAMS,
 
thereby determining the number of NAMs in the group of NAMs.

This application is a § 371 national stage of PCT InternationalApplication No. PCT/US2015/054981, filed Oct. 9, 2015, claiming benefitof U.S. Provisional Application No. 62/062,571, filed Oct. 10, 2014, thecontents of each of which are hereby incorporated by reference into theapplication.

Throughout this application, various publications are referenced,including referenced in parenthesis. Full citations for publicationsreferenced in parenthesis may be found listed at the end of thespecification immediately preceding the claims. The disclosures of allreferenced publications in their entireties are hereby incorporated byreference into this application in order to more fully describe thestate of the art to which this invention pertains.

REFERENCE TO SEQUENCE LISTING

This application incorporates-by-reference nucleotide sequences whichare present in the file named“180213_86434-A-PCT-US_Sequence_Listing_ADR.txt” which is 2 kilobytes insize, and which was created Feb. 13, 2018 in the IBM-PC machine format,having an operating system compatibility with MS-Windows, which iscontained in the text file filed Feb. 14, 2018.

BACKGROUND OF INVENTION

Despite improvements in DNA sequencing, many problems of interpretationarise when trying to count or assemble molecules (templates) that arelargely identical. Some problems in genomic analysis have remaineddifficult despite the development of high throughput sequencing methods.Many of these problems arise from the inability to distinguish identicaland nearly identical template sequences. Counting molecules of identicalcomposition in an RNA sequencing assay or the copy number of identicalstretches of DNA currently depend on quantitative methods that adjustimperfectly for the distortions of data caused by sample processing.

Moreover, when read lengths are short, determining the physicalconnection of distinguishable elements separated by long identicalstretches is difficult to impossible and limits the ability to phasesingle nucleotide variants (SNVs), identify transcript isoforms, andassemble through repetitive genomic regions.

There are several protocols in which a sequence of random nucleotides isappended to the template molecules before amplification and sequencing.This methodology has been applied under a variety of names to identifyPCR duplicates (McCloskey et al., 2007; Miner et al., 2004), improvecounting of DNA (Casbon et al., 2011; Fu et al., 2011) and RNA (Islam etal., 2014; Jabara et al., 2011; Kivioja et al., 2012) templates, andreduce sequence error (Hiatt et al., 2013; Kinde et al., 2011; Schmittet al., 2012). Each implementation has its own name for the randomnucleotide sequences, which can be referred to as varietal tags (Schmittet al., 2012). Each implementation also has its own shortcoming.

SUMMARY OF THE INVENTION

The present invention provides a method is provided for determining thenumber of nucleic acid molecules (NAMs) in a group of NAMs, comprising

-   -   i) obtaining an amplified and mutagenized group of NAMs that was        produced by        -   a) subjecting the group of NAMs to a chemical mutagenesis            which mutates only select nucleic acid bases in the group of            NAMs at a rate of 10% to 90% thus forming a group of            mutagenized NAMs (mNAMs), and        -   b) amplifying the group of mNAMs;    -   ii) obtaining sequences of the mNAMs in the group of amplified        mNAMs; and    -   iii) counting the number of different sequences obtained in        step (ii) to determine the number of unique mNAMs in the group        of mNAMS,        thereby determining the number of NAMs in the group of NAMs.

The present invention also provides a method for determining the numberof nucleic acid molecules (NAMs) in a group of NAMs, comprising

-   -   i) subjecting the group of NAMs to a chemical mutagenesis which        mutates only select nucleic acid bases in the group of NAMs at a        rate of 10% to 90%, to produce a group of mutagenized NAMs        (mNAMs);    -   ii) amplifying the group of mNAMs;    -   iii) sequencing the mNAMs in the group of amplified mNAMs to        obtain sequences of the mNAMs; and    -   iv) counting the number of different sequences obtained in        step (iii) to determine the number of unique mNAMs in the group        of mNAMs,        thereby determining the number of NAMs in the group of NAMs.

The present invention also provides a method for determining the numberof different sequences in a group of nucleic acid molecules (NAMs) thathave been mutagenized and then amplified comprising

-   -   a) obtaining the group of NAMs that have been mutagenized and        then amplified;    -   b) obtaining sequences of the mutagenized NAMs (mNAMs) in the        group of amplified mNAMs; and    -   c) counting the number of different sequences obtained in step        (ii),        thereby determining the number of different sequences in the        group of amplified mNAMs.

The present invention also provides a method for sequencing a nucleicacid molecule (NAM) that comprises two or more segments havingsubstantially the same sequence, and that has a length of more than onesequencing read, comprising

-   -   i) obtaining two or more copies of the NAM;    -   ii) subjecting each copy of the NAM in step (i) to a mutagenesis        that mutates only select nucleic acid bases in the NAMs at a        rate of 10% to 90% to produce mutated copies of the NAM (mcNAM);    -   iii) amplifying each of the mcNAMs;    -   iv) obtaining composite sequences of the mcNAMs that are        produced by assembling sequence reads of the amplified mcNAMs,        such that when taken together, span as much as possible of the        entire length of the NAM, by        -   a) aligning the sequence reads according to matching            mutation patterns in overlaps of the sequence reads, and        -   b) mapping the composite sequences,            thereby sequencing the NAM.

The present invention also provides a method for determining genomiccopy number information from genomic material, comprising,

-   -   i) obtaining segments of the genomic material; and    -   ii) determining the number of segments of the genomic material        according to the method of the claimed invention,        thereby determining genomic copy number information from genomic        material.

The present invention also provides a method for profiling RNAtranscripts, comprising

-   -   i) obtaining a group of RNA transcripts;    -   ii) determining the number of RNA transcripts in the group of        RNA transcripts according to the method of the claimed        invention; and    -   iii) determining the proportionate number of a plurality of RNA        transcripts having the same sequence to a second different        plurality of RNA transcripts that have the same sequence,        thereby determining RNA transcript profile.

The present invention also provides a method for determining allelicimbalance, comprising

-   -   i) obtaining copy number of a first allele;    -   ii) obtaining copy number of a second allele; and    -   iii) comparing the copy numbers obtained in steps (i) and (ii),        thereby determining allelic imbalance, wherein the copy number        in steps (i) and (ii) is obtained by the method of the claimed        invention.

The present invention also provides a method for determining genomeassembly, comprising

-   -   i) obtaining segments of a genome, wherein the segments span the        entire length of the genome;    -   ii) sequencing the segments of the genome according to the        method of the claimed invention;    -   iii) aligning the sequences obtained in step (ii) according to        matching mutation patterns in overlaps of the sequences; and    -   iv) mapping the sequences,        thereby assembling the genome.

The present invention also provides a method for determining haplotypeassembly, comprising

-   -   i) obtaining a group of alleles, wherein the alleles in the        group of alleles are located in the same chromosome;    -   ii) sequencing each allele in the group of alleles according to        the method of the claimed invention, and    -   iii) comparing the sequences obtained in step (ii),        thereby determining haplotype assembly.

The present invention also provides a kit for determining the number ofNAMs in a group of NAMs comprising:

-   -   a) a mutagen; and    -   b) instructions for performing mutagenesis resulting in        suboptimal mutagenesis,        wherein the mutagen is a bisulfite or a salt thereof, or a        deamination agent.

The present invention also provides a composition of matter comprising aplurality of mutagenized nucleic acid molecules (mNAMs), whereinselected mutable nucleic acid positions in the plurality of mutagenizedNAMs (mNAMs) are mutated at a rate of 10% to 90%.

The present invention also provides a composition of matter derived fromsequencing primers has the sequence:ACACTCTTTCCCTACACACGACGCTCTTCCGATC*T (SEQ ID NO:1), wherein thecytosines (C) are methylated, and wherein *T is a phosphorothioatedthymine.

The present invention also provides a composition of matter derived fromsequencing primers has the sequence:5Phos-GATCGGAAGAGCGGTTCAGCAGGAATGCCGA*G (SEQ ID NO:2), wherein thecytosines (C) are methylated, wherein *G is a phosphorothioated guanine,and wherein 5Phos is a 5′-phosphorylated, deoxyuridine-containinganchor-primer.

The present invention also provides a composition of matter comprisingtwo or more copies of a nucleic acid molecule (NAM) comprising two ormore segments having substantially the same sequence, and that has alength of more than one sequencing read, wherein each copy of the NAMhas a unique primer at its 5′ end and another unique primer at its 3′end, and is subjected to a mutagenesis that mutates each mutableposition in the NAMs at a rate of 10% to 90% to produce mutated copiesof the NAM (mcNAM), wherein the unique primers of each mcNAM lack anucleotide that is mutable by the mutagenesis.

The present invention also provides sequence information produced by asystem including one or more processing units which counts the number ofdifferent sequences obtained by a sequencer that processed the group ofamplified mNAMs in the method of the present invention, or the group ofmcNAMs of in the method of the present invention.

The present invention also provides a system including one or moreprocessing units which counts the number of different sequences obtainedby a sequencer that processed the group of amplified mNAMs of in themethod of the present invention, or the group of mcNAMs of in the methodof the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1. Enumerating Templates Over A Fixed Window

Illustration of the process of counting indistinguishable templatemolecules (panel A). In the first step, the molecules were mutated byrandom process, creating a unique mutation signature to each molecule(panel B). Cytidine deamination is also illustrated. Cytidinedeamination is a mutational method that converts cytidine (underlined)to a uridine (bold). Upon amplification, uridine becomes a thymidine(bold) (panel C). The sequenced nucleotide strings are aligned,aggregated by their mutational patterns (panel D), and the number of thedistinct patterns counted.

FIG. 2. Assembling Long Templates

Illustration of the process of connecting distinguishable markersseparated by a long span of indistinguishable sequence. In this example,each template has a unique pair of markers, or “end tags”, denoted bythe colored circle and square (panel A). Markers of the same color occuron the same template strands and are said to be “in phase”. Gray markson the templates show positions that may mutate. Each template wassubjected to a random mutation process (panel B) that converts some ofthe positions (converted positions shown in black). After amplificationand sequencing, each read is mapped to the reference template (panel C,top strand). Finally, by matching mutation patterns from overlappingreads, the phase of the original templates was recovered (panel D).

FIG. 3. Recovering Counts

The ability to recover the template count is a function of the windowsize, template length, flip rate, number of templates, and depth ofcoverage. Simulation results of template count estimation are shownunder a variety of conditions. Each panel has three plots, for windowsizes of 10, 20, and 30 bits. The x axis shows the true count from 2 to1,024 (log₂ scale) and the y axis shows the average estimated countdivided by the true count, or the proportion of templates recovered.Panel A simulates recovery when the template is one window long for arange of flip rates for infinite coverage. Panel B shows the resultsfrom one window template under finite coverage (1× to 5× reads pertemplate) for a fixed flip rate of 0.35. Panel C repeats the results ofpanel B for long templates comprised of 16 read lengths.

FIG. 4. Recovering Phase

Proper phasing of end tags separated by a long span of identicalsequence depends on window size, template length, flip rate, number oftemplates, and depth of coverage. In panels A and B, simulation resultsare shown for the recovery of phase for 32 templates across a spanranging from 2 to 1,024 read lengths for a 30-bit window. In panel A,infinite coverage was assumed and recovery was examined as a function offlip rate. Because coverage is exhaustive and the assembly graph is wellcharacterized, there are no errors and all phased results are correct.In panels B and C, the case of finite coverage was considered, fixingthe flip rate at 0.35 over a range of coverages from 2× to 14× pertemplate. In panel B, the number of template molecules was fixed at 32to explore the effect of template length on recovery of phase. In panelC, the template length was fixed to 32 read lengths to explore theeffect of the number of templates. Because not every read is observed, agreedy algorithm was used that may return exact matches that are correct(black) or incorrect (white).

FIG. 5. Greedy Path Assembly

To carry out the pattern assembly shown in FIG. 2, a graph was definedin which each read is vertex. Some reads contain their end tag (coloredcircles) and some do not (white circles). Two reads with an edge wereconnected if they agree on their overlap. The weight of an edge reflectsthe strength of that overlap. Panel A depicts the template informationassuming exhaustive coverage, drawing all distinct reads and the edgesbetween them. In panels B₁ and B₂, sample reads were finitely sampledfrom the templates at a depth of coverage of 4× and 8× per template,respectively. From this information the greedy algorithm was applied(panels C₁ and C₂), to select the best edges, shown in red. Whencoverage is low (panel C₁), some paths do not succeed in spanning thelength of the template and of those that do, three determine the correctphasing, and two are in error. Under higher coverage (panel C₂), allpaths span the template and all six are correctly phased.

FIG. 6. Partial Bisulfite Mutagenesis of a PhiX Template Genome

The experimental results from partial bisulfite mutagenesis of a phixtemplate genome. Panel A depicts the rate at which each base is observedover all the data for those positions with a coverage of at least 30reads. Panel B depicts the cumulative distribution of the conversionrate per read. Panel C depicts the correlation in flips for allcytosines in the targeted region. Panel D depicts panel C as ahistogram. For the best amplified position, partial conversion wasdetermined with a 60% flip rate randomly distributed throughout eachread.

FIG. 7. Conversion rate as a function of incubation time andtemperature. The datasets A3, A6, A9 and A45 are the 3, 6, 9 and 45minute conversions described herein.

FIG. 8. Subset of clustered reads showing mutational patterns and twoheterozygous positions in the sample. The panel on the left shows allthe positions in the fragment while the plot on the right shows onlycytosine (bit) positions. The white lines separate reads derived fromthe same initial template. Each cluster contains between 30 and 50reads. Black indicates a position where the read matches the referencegenome. The frequent gray squares are cytosines that have converted tothymine. The white and light grey streaks are linked heterozygousalleles which split according to mutation pattern. Sparse backgrounderrors are typically from the sequencer while the bands of error aretypically the result of PCR.

FIG. 9. Bit positions and bit patterns are uncorrelated. For the plot onthe left, pairs of positions (100,000) were sampled and the Fisher exactp-value computed to test the independence of bits by position. For theplot on the right, pairs of templates (100,000) were sampled and theFisher exact p-value computed to test the independence of bits within apattern. Both plots conform to the expected distribution.

FIG. 10. Comparison of template count distributions for fragments fromthe autosome and X chromosome. Since the sample is male, 2 to 1 ratiowas observed in the mean template counts. The histogram is the empiricaldistribution and the curve shows a negative binomial fit.

FIG. 11. Heterozygous allele counts by template demonstrate perfect fitto the binomial distribution. The plot on the left shows as a histogramthe counts for one allele. The curve shows the theoretical expectationof the count distribution assuming a binomial distribution for theallele at each locus assuming the given locus coverage. The plot on theright shows the Q-Q plot over all 6000 heterozygous positions observed.

FIG. 12. Heterozygous allele counts by read fail to fit binomialdistribution. As shown in FIG. 11, except the fit is quite poor. Farbetter to count templates than reads.

FIG. 13. Properties of the consensus sequences derived by clusteringreads with the same mutation pattern. For each consensus sequence base,the proportion of reads reporting the homozygous base was determined.These error rates are order of magnitude below sequencer error.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides a method for determining the number ofnucleic acid molecules (NAMs) in a group of NAMs, comprising

-   -   i) obtaining an amplified and mutagenized group of NAMs that was        produced by        -   a) subjecting the group of NAMs to a chemical mutagenesis            which mutates only select nucleic acid bases in the group of            NAMs at a rate of 10% to 90% thus forming a group of            mutagenized NAMs (mNAMs), and        -   b) amplifying the group of mNAMs;    -   ii) obtaining sequences of the mNAMs in the group of amplified        mNAMs; and    -   iii) counting the number of different sequences obtained in        step (ii) to determine the number of unique mNAMs in the group        of mNAMS,        thereby determining the number of NAMs in the group of NAMs.

The present invention provides a method for determining the number ofnucleic acid molecules (NAMs) in a group of NAMs, comprising

-   -   i) obtaining an amplified and mutagenized group of NAMs that was        produced by        -   a) subjecting the group of NAMs to a chemical mutagenesis            which mutates only select nucleic acid positions in the            group of NAMs at a rate of 10% to 90% thus forming a group            of mutagenized NAMs (mNAMs), and        -   b) amplifying the group of mNAMs;    -   ii) obtaining sequences of the mNAMs in the group of amplified        mNAMs; and    -   iii) counting the number of different sequences obtained in        step (ii) to determine the number of unique mNAMs in the group        of mNAMS,        thereby determining the number of NAMs in the group of NAMs.

In some embodiments, obtaining sequences comprises obtaining compositesequences produced by assembling sequence reads of the mNAMs by

-   -   a) aligning the sequence reads according to matching mutation        patterns in overlaps of the sequence reads, thereby obtaining        composite sequences, and    -   b) mapping the composite sequences; and        wherein counting the number of jointly overlapping different        composite sequences obtained.

In some embodiments, obtaining sequences in comprises obtainingcomposite sequences produced by assembling sequence reads of the mNAMsby

-   -   a) aligning the sequence reads according to matching mutation        patterns in overlaps of the sequence reads, thereby obtaining        composite sequences, and    -   b) mapping the composite sequences; and        wherein counting the number of different composite sequences        obtained.

The present invention also provides a method for determining the numberof nucleic acid molecules (NAMs) in a group of NAMs, comprising

-   -   i) subjecting the group of NAMs to a chemical mutagenesis which        mutates only select nucleic acid bases in the group of NAMs at a        rate of 10% to 90%, to produce a group of mutagenized NAMs        (mNAMs);    -   ii) amplifying the group of mNAMs;    -   iii) sequencing the mNAMs in the group of amplified mNAMs to        obtain sequences of the mNAMs; and    -   iv) counting the number of different sequences obtained in        step (iii) to determine the number of unique mNAMs in the group        of mNAMs,        thereby determining the number of NAMs in the group of NAMs.

The present invention also provides a method for determining the numberof nucleic acid molecules (NAMs) in a group of NAMs, comprising

-   -   i) subjecting the group of NAMs to a chemical mutagenesis which        mutates only select nucleic acid positions in the group of NAMs        at a rate of 10% to 90%, to produce a group of mutagenized NAMs        (mNAMs);    -   ii) amplifying the group of mNAMs;    -   iii) sequencing the mNAMs in the group of amplified mNAMs to        obtain sequences of the mNAMs; and    -   iv) counting the number of different sequences obtained in        step (iii) to determine the number of unique mNAMs in the group        of mNAMs,        thereby determining the number of NAMs in the group of NAMs.

In some embodiments, the sequencing comprises assembling sequence readsof the mNAMs into composite sequences by

-   -   a) aligning the sequence reads according to matching mutation        patterns in overlaps of the sequence reads, thereby obtaining        composite sequences, and    -   b) mapping the composite sequences; and        wherein counting the number of jointly overlapping different        composite sequences obtained.

In some embodiments, the sequencing comprises assembling sequence readsof the mNAMs into composite sequences by

-   -   a) aligning the sequence reads according to matching mutation        patterns in overlaps of the sequence reads, thereby obtaining        composite sequences, and    -   b) mapping the composite sequences; and        wherein counting the number of different composite sequences        obtained in step (iii).

In one or more embodiments, a sub-group of NAMs in the group of NAMs isdetermined to have substantially the same nucleotide sequence.

In one or more embodiments, the sub-group of NAMs is determined to havenucleotide sequences that are at least 95, 96, 97, 98, 99, 99.1, 99.2,99.3, 99.4, 99.5, 99.6, 99.7, 99.9, or 99.9% identical.

In one or more embodiments, the nucleotide sequences of a sub-group ofNAMs comprise a stretch of consecutive nucleotides having a sequencewhich includes at least two mutable positions and is i) identical to thesequence of a stretch of consecutive nucleotides within another NAMwithin the sub-group of NAMs, or ii) determined to have at least 95, 96,97, 98, 99, 99.1, 99.2, 99.3, 99.4, 99.5, 99.6, 99.7, 99.9, or 99.9%identical to the sequence of a stretch of consecutive nucleotides withinanother NAM within the sub-group of NAMs.

In one or more embodiments, the counting comprises counting the numberof different sequences that are determined to have substantially thesame sequence except for their mutable positions, thereby determiningthe number of NAMs in the group of NAMs that had substantially the samesequence.

In one or more embodiments, the counting comprises counting the numberof different sequences which lack substantially the same sequence in anystretch including at least two mutable positions, thereby determiningthe number of NAMs without substantially the same sequence in the groupof NAMs.

The present invention also provides a method for determining the numberof different sequences in a group of nucleic acid molecules (NAMs) thathave been mutagenized and then amplified comprising

-   -   a) obtaining the group of NAMs that have been mutagenized and        then amplified;    -   b) obtaining sequences of the mutagenized NAMs (mNAMs) in the        group of amplified mNAMs; and    -   c) counting the number of different sequences obtained in step        (ii),        thereby determining the number of different sequences in the        group of amplified mNAMs.

The present invention also provides a method for sequencing a nucleicacid molecule (NAM) that comprises two or more segments havingsubstantially the same sequence, and that has a length of more than onesequencing read, comprising

-   -   i) obtaining two or more copies of the NAM;    -   ii) subjecting each copy of the NAM in step (i) to a mutagenesis        that mutates only select nucleic acid positions in the NAMs at a        rate of 10% to 90% to produce mutated copies of the NAM (mcNAM);    -   iii) amplifying each of the mcNAMs;    -   iv) obtaining composite sequences of the mcNAMs that are        produced by assembling sequence reads of the amplified mcNAMs,        such that when taken together, span as much as possible of the        entire length of the NAM, by        -   a) aligning the sequence reads according to matching            mutation patterns in overlaps of the sequence reads, and        -   b) mapping the composite sequences,            thereby sequencing the NAM.

In some embodiments, a copy of the NAM is a partial copy of the NAM.

In some embodiments, a copy of the NAM has at least 50 bp of identicalor complementary sequence to the NAM.

In some embodiments, a copy of the NAM is a complete copy of the NAM.

The present invention also provides a method for sequencing a nucleicacid molecule (NAM) that comprises two or more segments havingsubstantially the same sequence, and that has a length of more than onesequencing read, comprising

-   -   i) obtaining two or more copies of the NAM;    -   ii) subjecting each copy of the NAM in step (i) to a mutagenesis        that mutates only select nucleic acid bases in the NAMs at a        rate of 10% to 90% to produce mutated copies of the NAM (mcNAM);    -   iii) amplifying each of the mcNAMs;    -   iv) obtaining composite sequences of the mcNAMs that are        produced by assembling sequence reads of the amplified mcNAMs,        such that when taken together, span as much as possible of the        entire length of the NAM, by        -   a) aligning the sequence reads according to matching            mutation patterns in overlaps of the sequence reads, and        -   b) mapping the composite sequences,            thereby sequencing the NAM.

The present invention also provides a method for sequencing a nucleicacid molecule (NAM) that comprises two or more segments havingsubstantially the same sequence, and that has a length of more than onesequencing read, comprising

-   -   i) obtaining two or more copies of the NAM;    -   ii) subjecting each copy of the NAM in step (i) to a chemical        mutagenesis that mutates only select nucleic acid bases in the        NAMs at a rate of 10% to 90% to produce mutated copies of the        NAM (mcNAM);    -   iii) amplifying each of the mcNAMs;    -   iv) obtaining composite sequences of the mcNAMs that are        produced by assembling sequence reads of the amplified mcNAMs,        such that when taken together, span as much as possible of the        entire length of the NAM, by        -   a) aligning the sequence reads according to matching            mutation patterns in overlaps of the sequence reads, and        -   b) mapping the composite sequences,            thereby sequencing the NAM.

In some embodiments, each of the two or more copies of the NAM has aunique primer at its 5′ end and another unique primer at its 3′ end.

In some embodiments, the unique primers of each mcNAM lack a nucleotidethat is mutable by the mutagenesis.

The present invention also provides a method for determining genomiccopy number information from genomic material, comprising,

-   -   i) obtaining segments of the genomic material; and    -   ii) determining the number of segments of the genomic material        according to the method of the present invention,        thereby determining genomic copy number information from genomic        material.

The present invention also provides a method for profiling RNAtranscripts, comprising

-   -   i) obtaining a group of RNA transcripts;    -   ii) determining the number of RNA transcripts in the group of        RNA transcripts according to the method of the claimed        invention; and    -   iii) determining the proportionate number of a plurality of RNA        transcripts having the same sequence to a second different        plurality of RNA transcripts that have the same sequence,        thereby determining RNA transcript profile.

The present invention also provides a method for determining allelicimbalance, comprising

-   -   i) obtaining copy number of a first allele;    -   ii) obtaining copy number of a second allele; and    -   iii) comparing the copy numbers obtained in steps (i) and (ii),        thereby determining allelic imbalance, wherein the copy number        in steps (i) and (ii) is obtained by the method of the present        invention.

The present invention also provides a method for determining genomeassembly, comprising

-   -   i) obtaining segments of a genome, wherein the segments span the        entire length of the genome;    -   ii) sequencing the segments of the genome according to the        method of the claimed invention;    -   iii) aligning the sequences obtained in step (ii) according to        matching mutation patterns in overlaps of the sequences; and    -   iv) mapping the sequences,        thereby assembling the genome.

The present invention also provides a method for determining haplotypeassembly, comprising

-   -   i) obtaining a group of alleles, wherein the alleles in the        group of alleles are located in the same chromosome;    -   ii) sequencing each allele in the group of alleles according to        the method of the claimed invention, and    -   iii) comparing the sequences obtained in step (ii),        thereby determining haplotype assembly.

In one or more embodiments, the rate of mutagenizing each mutableposition of the NAMs in the group of NAMs is 25% to 75%.

In one or more embodiments, the rate of mutagenizing each mutableposition of the NAMs in the group of NAMs is 40% to 60%.

In one or more embodiments, the rate of mutagenizing each mutableposition of the NAMs in the group of NAMs is 50%.

In one or more embodiments, the proportion of all bases mutated in eachmNAM is about 3% to 30%.

In one or more embodiments, the mutagenesis is by cytosine deamination.

In one or more embodiments, the mutagenesis is performed after bindingtemplate molecules to a bead or surface.

In one or more embodiments, biotinylated primers are attached totemplates.

In one or more embodiments, templates linked to biotinylated moietiesare attached to streptavidin beads.

In one or more embodiments, the mutagenesis further comprises beadsand/or varietal tags.

In some embodiments, the cytosine deamination is induced by a bisulfiteor a salt thereof.

In some embodiments, the cytosine deamination is induced by enzymology.

In some embodiments, the cytosine deamination is induced by anactivation-induced deaminase.

In one or more embodiments, the mutagenesis comprises contacting thegroup of NAMs with a depurination agent, transposase agent, or analkylating agent.

In one or more embodiments, each mutable position of the NAMs comprisesa cytosine (C).

In one or more embodiments, the cytosine (C) is mutated to a uracil (U)or a thymine (T).

In one or more embodiments, each NAM in the group of NAMs has a uniqueprimer at its 5′ end and another unique primer at its 3′ end.

In one or more embodiments, the primer comprises one or more methylatedcytosines.

In one or more embodiments, the primer comprises one or morephosphorothioated nucleotide bases.

In one or more embodiments, the primer further comprises a5′-phosphorylated, deoxyuridine-containing anchor-primer.

In one or more embodiments, the primer has the sequence:ACACTCTTTCCCTACACACGACGCTCTTCCGATCT (SEQ ID NO:3).

In one or more embodiments, the primer has the sequence:ACACTCTTTCCCTACACACGACGCTCTTCCGATC*T (SEQ ID NO:1), wherein thecytosines (C) are methylated, and wherein *T is a phosphorothioatedthymine.

In one or more embodiments, the cytosines (C) are methylated.

In one or more embodiments, the primer has the sequence:GATCGGAAGAGCGGTTCAGCAGGAATGCCGAG (SEQ ID NO:4)

In one or more embodiments, the primer further comprises a5′-phosphorylated, deoxyuridine-containing anchor-primer.

In one or more embodiments, the primer has the sequence: having thesequence: GATCGGAAGAGCGGTTCAGCAGGAATGCCGA*G (SEQ ID NO:2), wherein thecytosines (C) are methylated, wherein *G is a phosphorothioated guanine,and wherein 5Phos is a 5′-phosphorylated, deoxyuridine-containinganchor-primer.

In one or more embodiments, the cytosines (C) are methylated.

In one or more embodiments, the assembling further comprises aligningthe sequences according to unique primers at the 5′ and 3′ ends.

In one or more embodiments, the sequence of each unique primer comprisesa segment that is substantially the same sequence, and an amplificationprimer that is complementary to the shared sequence when amplifying themNAMs or copy thereof.

In some embodiments, amplification is performed using asequence-specific “wobble” primer.

In one or more embodiments, each unique primer comprises a unique tagsequence.

In one or more embodiments, the method further comprises the step oftagging each NAM or copy thereof.

In one or more embodiments, the tag lacks a nucleotide that is mutableby the mutagenesis.

In one or more embodiments, the NAM is within a mixture of DNA or

RNA extracted from a cell.

In some embodiments, the DNA or RNA extracted from the cell has beenfragmented.

In some embodiments, the DNA or RNA extracted from the cell has beenfragmented by mechanical shearing or one or more restriction enzymes.

In one or more embodiments, the one or more restriction enzyme is PstI.

In one or more embodiments, fragmentation occurs before amplifying.

In one or more embodiments, fragmentation occurs after amplifying.

In one or more embodiments, fragmentation occurs after mutagenesis.

In one or more embodiments, the method of the claimed invention furthercomprises subjecting the fragmented DNA or RNA to end-repair.

In one or more embodiments, the method of the claimed invention furthercomprises subjecting the fragmented DNA or RNA to adenylation.

In one or more embodiments, the method of the claimed invention furthercomprises subjecting the fragmented DNA or RNA to ligation withmethyl-cytosine adaptors, wherein the methyl-cytosine adaptors arebisulfite resistant sequencing adaptors.

In one or more embodiments, the NAM is a DNA molecule.

In some embodiments, the DNA molecule is a fragment of genomic DNA.

In one or more embodiments, the DNA molecule is a cDNA molecule.

In one or more embodiments, the NAM is an RNA molecule.

In one or more embodiments, the RNA molecule is an mRNA molecule.

In one or more embodiments, the RNA molecule is a viral RNA molecule.

In one or more embodiments, the NAM is an RNA molecule derived from oneor more cell lines.

In one or more embodiments, the method of the claimed invention furthercomprises reverse transcription of the NAM.

In one or more embodiments, the reverse transcription is with poly-T andmethyl-cytosines, wherein the methyl-cytosines are resistant tobisulfite mutation.

In one or more embodiments, chemical mutagenesis occurs prior to reversetranscription.

In one or more embodiments, one or more NAMs in the group of NAMs has alength of one sequencing read length.

In one or more embodiments, one or more NAMs in the group of NAMs has alength of two or more sequencing read lengths.

In one or more embodiments, the sequencing read length is 2, 3, 4, 5, 6,7, 8, 9, 10, or 10-3000 sequencing read lengths.

In one or more embodiments, the number of NAMs in the group of NAMs isabout 2, 3, 4, 5, 6, 7, 8, 9, 10, or 10-10000.

In one or more embodiments, the number of NAMs in the group of NAMs isgreater than 10000, then diluting the group of NAMs.

In one or more embodiments, diluting the group of NAMs to comprise 1000or more NAMs in the group of NAMs.

In one or more embodiments, the amplifying is by short-range orlong-range polymerase chain reaction (PCR).

In one or more embodiments, the mutagen of the mutagenesis is diluted.

In one or more embodiments, the group of NAMs is incubated with amutagen at an incubation temperature of about 70 to 78 degrees Celsius.

In one or more embodiments, the incubation temperature is about 73degrees Celsius.

In one or more embodiments, the group of NAMs is incubated with amutagen at an incubation time of about 3 to 45 minutes.

In one or more embodiments, the group of NAMs is incubated with amutagen at an incubation time of about 5 to 20 minutes.

In one or more embodiments, the incubation time is about 3, 6, or 9minutes.

In one or more embodiments, the incubation time is about 10 minutes.

The present invention also provides a kit for determining the number ofNAMs in a group of NAMs comprising:

-   -   a) a mutagen; and    -   b) instructions for performing mutagenesis resulting in        suboptimal mutagenesis,        wherein the mutagen is a bisulfite or a salt thereof, or a        deamination agent.

In some embodiments, the bisulfite or salt thereof is NaHSO₃.

In one or more embodiments, the mutagen induces cytosine deamination.

In one or more embodiments, the cytosine deamination is by enzymology.

In one or more embodiments, the mutagen is diluted.

In one or more embodiments, the kit of the present invention furthercomprises a plurality of unique primers including:

-   -   a) a plurality of substantially unique primers suitable for        ligation to the 5′ of a NAM; and    -   b) a plurality of substantially unique primers suitable for        ligation to the 3′ of a NAM.        wherein the substantially unique primers comprise substantially        unique tags.

In one or more embodiments, the kit of the present invention furthercomprises a DNA polymerase having 3′-5′ proofreading activity.

In one or more embodiments, the plurality of substantially uniqueprimers comprises 10^(n) primers, wherein n is an integer from 2 to 9.

In one or more embodiments, the substantially unique tags are at least 6nucleotides long.

In one or more embodiments, the substantially unique tags are at least15 nucleotides long.

In one or more embodiments, the substantially unique primers comprisesets of substantially unique primers having shared sample tags.

In one or more embodiments, the sample tags are at least 2 or 4nucleotides long.

In one or more embodiments, the sequence of the substantially unique tagis not altered by the mutagen.

In one or more embodiments, the kit of the present invention furthercomprises a primer wherein the cytosines (C) are methylated.

In one or more embodiments, the methylated primer having the sequence:ACACTCTTTCCCTACACACGACGCTCTTCCGATC*T (SEQ ID NO:1), wherein thecytosines (C) are methylated, and wherein *T is a phosphorothioatedthymine.

In one or more embodiments, the methylated primer having the sequence:5Phos-GATCGGAAGAGCGGTTCAGCAGGAATGCCGA*G (SEQ ID NO:2), wherein thecytosines (C) are methylated, wherein *G is a phosphorothioated guanine,and wherein 5Phos is a 5′-phosphorylated, deoxyuridine-containinganchor-primer.

The present invention also provides a composition of matter comprising aplurality of mutagenized nucleic acid molecules (mNAMs), whereinselected mutable nucleic acid positions in the plurality of mutagenizedNAMs (mNAMs) are mutated at a rate of 10% to 90%.

In one or more embodiments, each mutable position is mutated at a rateof 25% to 75%.

In one or more embodiments, each mutable position is mutated at a rateof 40% to 60%.

In one or more embodiments, each mutable position is mutated at a rateof 50%.

In one or more embodiments, each mutable nucleic acid base is mutated ata rate of 25% to 75%.

In one or more embodiments, each mutable nucleic acid base is mutated ata rate of 40% to 60%.

In one or more embodiments, each mutable nucleic acid base is mutated ata rate of 50%.

In one or more embodiments, the proportion of all nucleic acids mutatedin each mNAM is about 3% to 30%.

In one or more embodiments, the mutable nucleic acid position is acytosine position of the mNAMs and the mutagenesis is deamination of thecytosine.

In one or more embodiments, the mutable nucleic acid base is a cytosinebase of the mNAMs and the mutagenesis is deamination of the cytosine.

In one or more embodiments, the deamination of the cytosine is inducedby a bisulfite or a salt thereof.

In one or more embodiments, the cytosine deamination of the cytosine isinduced by enzymology.

In one or more embodiments, the cytosine deamination of the cytosine isinduced by an activation-induced deaminase.

In one or more embodiments, the mutable nucleic acid position ismutagenized by contacting the group of NAMs with a depurination agent,transposase agent, or an alkylating agent.

In one or more embodiments, each mutable position of the NAMs comprisesa cytosine (C).

In one or more embodiments, the cytosine (C) is mutated to a uracil (U)or a thymine (T).

In one or more embodiments, each NAM in the plurality of NAMs has aunique primer at its 5′ end and another unique primer at its 3′ end.

In one or more embodiments, the primer comprises one or more methylatedcytosines.

In one or more embodiments, the primer comprises one or morephosphorothioated nucleotide bases.

In one or more embodiments, the primer further comprises a5′-phosphorylated, deoxyuridine-containing anchor-primer.

In one or more embodiments, the primer has the sequence:ACACTCTTTCCCTACACACGACGCTCTTCCGATCT (SEQ ID NO:3).

In one or more embodiments, the plurality of mNAMS bearing a primerwherein the sequence of the primer is:ACACTCTTTCCCTACACACGACGCTCTTCCGATC*T (SEQ ID NO:1), and wherein *T is aphosphorothioated thymine.

In one or more embodiments, the cytosines (C) are methylated.

In one or more embodiments, all the cytosines (C) are methylated.

In one or more embodiments, the primer has the sequence:GATCGGAAGAGCGGTTCAGCAGGAATGCCGAG (SEQ ID NO:4).

In one or more embodiments, the primer further comprises a5′-phosphorylated, deoxyuridine-containing anchor-primer.

In one or more embodiments, the plurality of mNAMS bearing a primerwherein the sequence of the primer is:5Phos-GATCGGAAGAGCGGTTCAGCAGGAATGCCGA*G (SEQ ID NO:2), wherein *G is aphosphorothioated guanine, and wherein 5Phos is a 5′-phosphorylated,deoxyuridine-containing anchor-primer.

In one or more embodiments, the cytosines (C) are methylated.

The present invention also provides a composition of matter derived fromsequencing primers has the sequence:ACACTCTTTCCCTACACACGACGCTCTTCCGATC*T (SEQ ID NO:1), wherein thecytosines (C) are methylated, and wherein *T is a phosphorothioatedthymine.

The present invention also provides a composition of matter derived fromsequencing primers has the sequence:5Phos-GATCGGAAGAGCGGTTCAGCAGGAATGCCGA*G (SEQ ID NO:2), wherein thecytosines (C) are methylated, wherein *G is a phosphorothioated guanine,and wherein 5Phos is a 5′-phosphorylated, deoxyuridine-containinganchor-primer.

The present invention also provides a composition of matter comprisingtwo or more copies of a nucleic acid molecule (NAM) comprising two ormore segments having substantially the same sequence, and that has alength of more than one sequencing read, wherein each copy of the NAMhas a unique primer at its 5′ end and another unique primer at its 3′end, and is subjected to a mutagenesis that mutates each mutableposition in the NAMs at a rate of 10% to 90% to produce mutated copiesof the NAM (mcNAM), wherein the unique primers of each mcNAM lack anucleotide that is mutable by the mutagenesis.

The present invention also provides sequence information produced by asystem including one or more processing units which counts the number ofdifferent sequences obtained by a sequencer that processed the group ofamplified mNAMs in the method of the claimed invention, or the group ofmcNAMs of in the method of the claimed invention.

The present invention also provides a system including one or moreprocessing units which counts the number of different sequences obtainedby a sequencer that processed the group of amplified mNAMs of in themethod of the claimed invention, or the group of mcNAMs of in the methodof the claimed invention.

The present invention also provides a method for sequencing a nucleicacid molecule (NAM) that comprises two or more segments havingsubstantially the same sequence, and that has a length of more than onesequencing read, comprising

-   -   i) obtaining two or more copies of the NAM;    -   ii) subjecting each copy of the NAM in step (i) to a mutagenesis        that mutates only select nucleic acid positions in the NAMs at a        rate of 10% to 90% to produce mutated copies of the NAM (mcNAM);    -   iii) amplifying each of the mcNAMs;    -   iv) obtaining composite sequences of the mcNAMs that are        produced by assembling sequence reads of the amplified mcNAMs,        such that when taken together, span as much as possible of the        entire length of the NAM, by de novo assembly,        thereby sequencing the NAM.

The present invention also provides a method for distinguishing betweenbenign and malignantly transformed cells by detecting one or more singlenucleotide polymorphisms (SNPs) in a sample from a subject and areference sample from a control subject comprising a method of theclaimed invention.

The present invention also provides a method for distinguishing betweenbenign and malignantly transformed cells by detecting one or more singlenucleotide polymorphisms (SNPs) in a first and second sample from asubject comprising a method of the claimed invention.

The present invention also provides a method for determining thepresence of tumor cells in a sample by comparing a sample from a subjectand a reference sample from a control subject comprising a method of theclaimed invention.

The present invention also provides a method for determining thepresence of tumor cells in a sample by comparing a first and secondsample from a subject comprising a method of the claimed invention.

The present invention also provides a method for quantifying tumor cellsin a sample by comparing a sample from a subject and a reference samplefrom a control subject comprising a method of the claimed invention.

The present invention also provides a method for quantifying tumor cellsin a sample by comparing a sample from a first and second sample from asubject comprising a method of the claimed invention.

The present invention also provides a method for detecting one or morerare mutations by comparing a sample from a subject and a referencesample from a control subject comprising a method of the claimedinvention.

The present invention also provides a method for detecting one or morerare mutations by comparing a sample from a first and second sample froma subject comprising a method of the claimed invention.

In one or more embodiments, the sample is a blood sample, plasma sample,serum sample, tissue sample, or cell sample.

In one or more embodiments, the tissue sample is from a tumor mass,surgically removed tumor mass, or margins of a surgically removed tumormass.

The present invention also provides a method for detecting one or morerare mutations in a cell-free or substantially cell-free samplecomprising a method of the claimed invention.

The present invention also provides a method for determining whether afetus has at least one or more rare mutations in a cell-free orsubstantially cell-free sample comprising a method of the claimedinvention

In one or more embodiments, the sample is a maternal sample.

In one or more embodiments, the maternal sample is obtained from amember selected from: maternal blood, maternal plasma and maternalserum.

Each embodiment disclosed herein is contemplated as being applicable toeach of the other disclosed embodiments. Thus, all combinations of thevarious elements described herein are within the scope of the invention.

It is understood that where a parameter range is provided, all integerswithin that range, and tenths thereof, are also provided by theinvention. For example, “70 to 78 degrees Celsius” is a disclosure of70.0 degrees Celsius, 70.1 degrees Celsius, 70.2 degrees Celsius, 70.3degrees Celsius, 70.4 degrees Celsius, 70.5 degrees Celsius etc. up to78.0 degrees Celsius.

Terms

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by a person of ordinaryskill in the art to which this invention belongs.

As used herein, and unless stated otherwise or required otherwise bycontext, each of the following terms shall have the definition set forthbelow.

As used herein, “about” in the context of a numerical value or rangemeans ±10% of the numerical value or range recited or claimed, unlessthe context requires a more limited range.

The terms “nucleic acid molecule” and “sequence” are not usedinterchangeably herein. A “sequence” refers to the sequence informationof a “nucleic acid molecule”.

As used herein “mutable position” refers to the position of a nucleicacid that is susceptible to a given type of chemical mutagenesis.

As used herein “determining the number” refers to determining the lowerbound number.

As used herein “X %” with respect to mutation rate, refers to theprobability percentage of mutagenesis per mutable position of themultiple mutable positions that are present in a plurality of nucleicacid molecules. Thus, 25% mutation rate means a 25% probability ofmutagenesis.

The terms “template”, “nucleic acid”, and “nucleic acid molecule”, areused interchangeably herein, and each refers to a polymer ofdeoxyribonucleotides and/or ribonucleotides. “Nucleic acid” shall meanany nucleic acid, including, without limitation, DNA, RNA and hybridsthereof. The nucleic acid bases that form nucleic acid molecules can bethe bases A, C, G, T and U, as well as derivatives thereof.

As used herein “contig” and “continguous” refers to a set of overlappingsequence or sequence reads.

As used herein, the term “amplifying” refers to the process ofsynthesizing nucleic acid molecules that are complementary to one orboth strands of a template nucleic acid. Amplifying a nucleic acidmolecule typically includes denaturing the template nucleic acid,annealing primers to the template nucleic acid at a temperature that isbelow the melting temperatures of the primers, and enzymaticallyelongating from the primers to generate an amplification product. Thedenaturing, annealing and elongating steps each can be performed once.Generally, however, the denaturing, annealing and elongating steps areperformed multiple times (e.g., polymerase chain reaction (PCR)) suchthat the amount of amplification product is increasing, often timesexponentially, although exponential amplification is not required by thepresent methods. Amplification typically requires the presence ofdeoxyribonucleoside triphosphates, a DNA polymerase enzyme and anappropriate buffer and/or co-factors for optimal activity of thepolymerase enzyme. The term “amplified nucleic acid molecule” refers tothe nucleic acid sequences, which are produced from the amplifyingprocess as defined herein.

As used herein, the term “bisulfite mutagenesis” refers to themutagenesis of nucleic acid with a reagent used for the bisulfiteconversion of cytosine to uracil. Examples of bisulfite conversionreagents include but are not limited to treatment with a bisulfite, adisulfite or a hydrogensulfite compound.

As used herein, the term “mapping” refers to identifying a location on agenome or cDNA library that has a sequence which is substantiallyidentical to or substantially fully complementary. The nucleic acidmolecule may be, but is not limited to the following: a segment ofgenomic material, a cDNA, a mRNA, or a segment of a cDNA.

As used herein, the term “methylation” refers to the covalent attachmentof a methyl group at the C5-position of the nucleotide base cytosine.The term “methylation state” or refers to the presence or absence of5-methyl-cytosine (“5-Me”) at one or a plurality of CpG dinucleotideswithin a DNA sequence. A methylation site is a sequence of contiguouslinked nucleotides that is recognized and methylated by a sequencespecific methylase. A methylase is an enzyme that methylates (i. e.,covalently attaches a methyl group) one or more nucleotides at amethylation site.

As used herein, the term “read” or “sequence read” refers to thenucleotide or base sequence information of a nucleic acid that has beengenerated by any sequencing method. A read therefore corresponds to thesequence information obtained from one strand of a nucleic acidfragment. For example, a DNA fragment where sequence has been generatedfrom one strand in a single reaction will result in a single read.However, multiple reads for the same DNA strand can be generated wheremultiple copies of that DNA fragment exist in a sequencing project orwhere the strand has been sequenced multiple times. A read thereforecorresponds to the purine or pyrimidine base calls or sequencedeterminations of a particular sequencing reaction.

As used herein, the terms “sequencing”, “obtaining a sequence” or“obtaining sequences” refer to nucleotide sequence information that issufficient to identify or characterize the nucleic acid molecule, andcould be the full length or only partial sequence information for thenucleic acid molecule.

As used herein, the term “reference genome” refers to a genome of thesame species as that being analyzed for which genome the sequenceinformation is known.

As used herein, the term “region of the genome” refers to a continuousgenomic sequence comprising multiple discrete locations.

As used herein, the term “sample tag” refers to a nucleic acid having asequence no greater than 1000 nucleotides and no less than two that maybe covalently attached to each member of a plurality of tagged nucleicacid molecules or tagged reagent molecules. A “sample tag” may comprisepart of a “tag.”

As used herein, the term “segments of genomic material” refers to thenucleic acid molecules resulting from fragmentation of genomic DNA.

As used herein, “substantially the same” sequences have at least about80% sequence identity or complementarity, respectively, to a nucleotidesequence. Substantially the same sequences or may have at least about95%, 96%, 97%, 98%, 99% or 100% sequence identity or complementarity,respectively.

As used herein, the term “substantially unique primers” refers to aplurality of primers, wherein each primer comprises a tag, and whereinat least 50% of the tags of the plurality of primers are unique.Preferably, the tags are at least 60%, 70%, 80%, 90%, or 100% uniquetags.

As used herein, the term “substantially unique tags” refers to tags in aplurality of tags, wherein at least 50% of the tags of the plurality areunique to the plurality of tags. Preferably, substantially unique tagswill be at least 60%, 70%, 80%, 90%, or 100% unique tags.

As used herein, the term “tag” refers to a nucleic acid having asequence no greater than 1000 nucleotides and no less than two that maybe covalently attached to a nucleic acid molecule or reagent molecule. Atag may comprise a part of an adaptor or a primer.

As used herein, a “tagged nucleic acid molecule” refers to a nucleicacid molecule which is covalently attached to a “tag.”

Where a range of values is provided, it is understood that eachintervening value, to the tenth of the unit of the lower limit unlessthe context clearly dictates otherwise, between the upper and lowerlimit of that range, and any other stated or intervening value in thatstated range, is encompassed within the invention. The upper and lowerlimits of these smaller ranges may independently be included in thesmaller ranges, and are also encompassed within the invention, subjectto any specifically excluded limit in the stated range. Where the statedrange includes one or both of the limits, ranges excluding either orboth of those included limits are also included in the invention.

As used herein, the term “wobble base pairing” with regard to twocomplementary nucleic acid sequences refers to the base pairing of G touracil U rather than C, when one or both of the nucleic acid strandscontains the ribonucleobase U.

As used herein, the term “substantially fully complementary” with regardto a sequence refers to the reverse complement of the sequence allowingfor both Watson-Crick base pairing and wobble base pairing, whereby Gpairs with either C or U, and A pairs with either U or T. A sequence maybe substantially complementary to the entire length of another sequence,or it may be substantially complementary to a specified portion orlength of another sequence. One of skill in the art will recognize thatthe U may be present in RNA, and that T may be present in DNA.Therefore, a U within an RNA sequence may pair with A or G in either anRNA sequence or a DNA sequence, while an A within either of a RNA or DNAsequence may pair with a U in a RNA sequence or T in a DNA sequence.

As used herein, a “wobble” primer is a set of primers where the sequenceat mutable positions is equally likely to match the original base or themutated base.

All publications and other references mentioned herein are incorporatedby reference in their entirety, as if each individual publication orreference were specifically and individually indicated to beincorporated by reference. Publications and references cited herein arenot admitted to be prior art.

This invention will be better understood by reference to theExperimental Details which follow, but those skilled in the art willreadily appreciate that the specific experiments detailed are onlyillustrative of the invention as defined in the claims which followthereafter.

EXPERIMENTAL DETAILS

Examples are provided below to facilitate a more complete understandingof the invention. The following examples illustrate the exemplary modesof making and practicing the invention. However, the scope of theinvention is not limited to specific embodiments disclosed in theseExamples, which are for purposes of illustration only.

Methods—Counting Templates (Single Read Length)

To enable a wide range of simulations, a library of Python programs wasdeveloped, (Example 11), to simulate mutation, sequencing, counting andassembly of distinct templates under the assumptions of error-freesequencing, perfect mapping, and uniformity of mutation sites, mutationrate, sequence coverage, and DNA amplification. Our ability to recovertemplate count and assembly depends on the depth of read sampling,typically called “coverage”. Coverage usually means the average numberof reads overlapping a position in the reference genome, however herein,coverage means the average read depth over a position per template.

To measure template count over a single fixed read length for Ttemplates, mutation was simulated to generate T random template patternswith R bits and a flip rate of p. To measure recovery under infinitecoverage, the number of distinguishable mutation patterns observed wascounted (Example code in Section Z). To measure recovery under finitecoverage, reads were generated by sampling a fixed number of patternswith replacement from the pool before counting distinguishable patterns(Example code in Section Z). Amplification distortion could be modeledby altering the sampling procedure; however, results are restricted tothe case of uniform sampling. Simulations explore mutation rates in therange of 0.05 to 0.35, template counts in the range of 2 to 1,024, andread lengths with 10, 20, and 30 mutable bits. For each condition, onehundred simulations were performed and record the average recovery underinfinite coverage (FIG. 3, panel A). For each of those hundredsimulations, one hundred finite samplings were performed for eachcoverage level of 1× to 5× per template and the mean count observed wasrecorded. Shown in FIG. 3, panel B are the results for the mutation rateof 0.35.

The first class of applications focuses on the general problem ofdetermining absolute template count. This is important for determiningthe copy number of genomic DNA, measuring mRNA expression levels,quantifying allele bias, and detecting somatic mutations. To obtain anaccurate count, the protocol requires mutagenesis prior toamplification. Amplification could be either short- or long-range PCRbut must occur before fragmentation if needed for library preparation.The number of possible mutation patterns should exceed the templatenumber to obtain the most accurate count. Hence, cases were onlyconsidered where the absolute template count is below the low thousands,and save the other cases for Discussion.

Example 1. Counting Templates (Single Read Length)

The simplest formulation is the case where the templates span a singlefixed read length and exhaustive coverage (FIG. 3, panel A). In thiscase, the estimate of template count is merely the number ofdistinguishable mutation patterns observed.

The number of possible patterns depends on the number of bits per read,and the probability of observing a given pattern depends also on theflip rate. The optimal flip rate for generating distinct mutationalpatterns is 0.5, wherein every pattern is equally likely. However for awindow of at least 20 bits, corresponding to a read length of 80 basepairs, a rate of 0.25 is still virtually perfect for template counts inthe thousands. Similar efficacy is obtained at a flip rate of 0.15 for a30-bit window. Templates numbering in the thousands are adequate forgenome copy number determination or single-cell transcript profiling. InSection Z, example code is provided to simulate performance under avariety of conditions.

The recovery of template count was also demonstrated subject to varyingdepth of coverage for a fixed flip rate of 0.35 (FIG. 3, panel B). Foreach simulation, T initial template molecules were mutated, creating apool of patterns. At a coverage level of c, c×T patterns were selectedwith replacement from the pool and record the number of distinctpatterns. For high coverage, on the order of 4× per original templatemolecule, recovery of count is nearly perfect. At lower coverage thecount is underestimated, the inevitable consequence of undersampling.These simulations assume uniform PCR amplification and provide an upperbound on performance.

Methods—Counting Templates (Multiple Read Lengths)

When the template length exceeds a single window, a few simple graphformalisms are used. Two reads are said to “conflict” if they map tooverlapping positions and their overlaps fail to agree. A “compatibleread graph” was defined to be one in which each distinguishable read istreated as a “vertex” and two vertices are “compatible” and connected byan “edge” if the reads could have originated from the same template. Inthe case of finite coverage, two reads are compatible if they do notconflict. In the case of infinite coverage, there is a strongerconstraint: two reads at a distance d mutable bits apart are compatibleif all d−1 were observed to be distinct, non-conflicting, intermediatereads (Example code in Section Z). All edges inherit a direction fromthe orientation of the reference template such that each vertex hasin-edges extending from one end of the template and out-edges extendingtoward the other end.

A path in this graph represents a possible partial assembly of aninitial template pattern. Consequently, determining the minimum numberof templates needed to explain all of the reads is achieved by findingthe minimum number of paths such that every vertex in the graph isincluded in at least one path. This is known as the minimum vertex coverand in general is an nondeterministic polynomial time hard (NP-hard)problem. However, under the assumption of perfect read mapping to alinear genome, our graph is not only directed, but also acyclic. ByDilworth's theorem the minimum number of covering paths is equivalent tothe maximum number of elements in an antichain (Dilworth, 1950). Inother words, the minimum number of templates needed to explain the readsis equal to the maximum number of reads that are pairwise incompatible.Using König's theorem König, 1931), this problem was solved by finding amaximal matching in a new bipartite graph constructed by splitting eachvertex in two (an “in” vertex that receives in-edges and an “out” vertexthat receives out-edges). Finding a maximal matching in a bipartitegraph is then solvable in polynomial time by the Hoperoft-Karp algorithm(Hoperoft and Karp, 1973).

As in the case of a fixed read length, mutation was simulated togenerate a pool of template patterns. For a finite coverage level of c,read length R, and T initial templates of length L, c×T×(L/R) reads weregenerated by drawing uniformly with replacement from the set oftemplates and read start positions. These reads were used to build acompatible read graph, convert to a bipartite graph, and applyHoperoft-Karp to find a maximal matching.

In the case of finite coverage, all reads that do not overlap are bydefinition compatible. This implies that the maximal antichain(comprised of reads that are all pairwise incompatible) must becomprised of reads all in which all overlap and therefore all startwithin a single interval of length R. This fact permits a significantcomputational simplification: instead of computing the maximal antichainfor the entire graph, smaller subgraphs of reads can be restricted whosestart positions are contained in the interval [nR, (n+2)R] for n=0, . .. , (L/R)−1; identify a maximal antichain over each interval; andcompute the maximum size over all intervals (Example code in Section Z).For a fixed mutation rate of 0.35, emplate counts were simulated in therange of 2 to 1,024, reads of 10, 20, and 30 bits, templates that are 16read lengths long, and a coverage of 1× to 5×. For each condition, 100simulations were performed and the mean number of templates recoveredwere reported.

Example 2. Counting Templates (Multiple Read Lengths)

More often, the length of the template will exceed a single read length.In this case, there was no fixed window over which to count distinctmutational patterns. So instead of looking for the number of unique (andhence incompatible) patterns over a fixed window, a maximal set ofpairwise incompatible reads was determined. This problem can beprecisely stated in the language of graph theory. More specifically,reads were treated as vertices, and connected by directed “compatible”edges whenever two reads can derive from the same template. Thedirection of the edge is determined by the orientation of the referencetemplate. The result is a directed acyclic graph. By Dilworth's theorem(Dilworth, 1950), the size of the largest set of pairwise incompatiblereads is the same as the minimum number of paths covering all verticesof this graph. By König's theorem (König, 1931) this is a problem ofmaximal matching in a bipartite graph, which is computationallytractable and solvable in polynomial time with the Hoperoft-Karpalgorithm (Hoperoft and Karp, 1973). In FIG. 3 panel C, the results areshown for a fixed mutation rate of 0.35, a template that spans 16 readlengths, for depths of coverage ranging from 1× to 5× per template, andfor template counts ranging from 2 to 1,024. Under similar conditions offlip rate and coverage, counting over long templates is comparable toperformance in counting over a fixed window, as described above.

The simulations of this section provide guidelines for (i) genome widecopy number determination, (ii) transcript profiling, and (iii)determining allelic ratios. To determine copy number, the ratio of countwas measured for a given locus to the median count over the remainder ofthe genome. For transcript profiling, the proportionate counts of eachgene transcript were measured. To determine allelic imbalance, the ratioof counts was measured from templates distinguishable by at least oneSNV. In the context of RNA, this also enables observation of biasedallele expression resulting from chromosome inactivation, imprinting andthe like.

Methods—Assembling Templates

In the counting problem, a conservative estimate was established for theinitial number of templates by allowing all compatible edges betweenreads. For the assembly problem, however, high probability assemblieswere to be established. Consequently, the compatible edge graph wasrestricted to a sub-graph composed of the best edges. When coverage isexhaustive, a very precise definition of best edges can be formulated.Two R-bit reads were joined by an edge if they overlap and agree for R−1bits. Consequently, two tags of a template are exactly matched if andonly if every such R−1 bit string across its span has a unique pattern.When the condition is not met—for example due to too many templatesand/or too few flipped bits—connected end tags that were not exactlymatched were found (Example code in Section Z).

When coverage is finitely sampled, there are many ways to select thebest edges for a subgraph (FIG. 5). A simple scoring method was appliedthat assigns a weight to an edge that reflects how unlikely it is thatthe reads agree on their mutation patterns by chance. Given two readsthat overlap and agree on a window of size M with K bits flipped, theedge was weighted by −log(p^(K)(1−p)^(M-K)) where p is the flip rate.All of the edges in order of decreasing weight were iterated through. Anedge e from A to B is selected for inclusion in the subgraph if no edgefrom A or to B has already been selected that has a weight strictlygreater than e. This procedure was carried out for each simulation andextract from the resultant subgraph all exact matches. Unlike the casefor exhaustive coverage, exact matches may be incorrect. Because this isa simulation, the truth was known and the number of exact matches thatare correct and incorrect was recorded. Whether the exact path is alsotrue was also recorded (Example code in Section Z).

Example 3. Assembling Templates

The second class of applications is to correctly assemble reads by theirmutation patterns in order to recover the proper end-to-end sequence ofnearly identical templates, desired when determining haplotype phasingor enumerating transcript isoforms. Long templates each tagged uniquelyat both ends were considered to simulate the more general task ofdetermining how many initial templates can be correctly assembled fromend to end from the mutation pattern alone (FIG. 2).

Following mutagenesis, amplification, fragmentation, and sequencing,reads were connected with overlapping mutation signatures to assemble apath from one tag to the other. Whereas in the previous application, allcompatible edges between reads were allowed, for this problem a subgraphwas built with only the “best edges” between overlapping reads. A pairof tags is “exactly matched” if there is a path in the subgraph thatconnects them and neither tag is connected to another tag. Such a pathis called an “exact path.” If two tags originate from the same template,they are a “true match.” A “true path” is an exact path for which everyread originates from the same initial template.

Determining performance for the general task provides a lower bound onperformance for other applications, because if there is an exact paththat is also true, then all sequence information for that template wascorrectly observed. This includes haplotype phasing in the case ofgenomic data and transcript structure in the case of RNA profiling. Infact, these two applications are less demanding than the general taskbecause there will only be a few template varieties and each templatevariety provides additional sequence information for distinguishingthem.

The case of exhaustive coverage was considered to establish the bestperformance that can expect for a given set of conditions. In this case,the best edges are naturally defined as the set of compatible edges thatoverlap for all but one bit. In FIG. 4 panel A, the effect of flip rateon template assembly was explored. At a flip rate of 0.35 and 30 bitsper read, performance on 32 template molecules is nearly perfect foreven the longest span tested, 2¹⁰ read lengths, in excess of 100kilobases.

With exhaustive coverage, best edges are unambiguously defined and havethe property that exactly matched pairs and exact paths are also true.However, for finite coverage, there is no natural definition of bestedges and exactly matched pairs and exact paths may not be true. Forfinite coverage, weights to edges were assigned by likelihood and agreedy algorithm was used to select the best subgraph (detailed aboveand illustrated in FIG. 5).

FIG. 4 panel B explores the effect of coverage (2× to 14× per template)on recovery of exact matches as a function of template length (2 to1,024 read lengths), for 32 templates, a 30 bit read length, and a fliprate of 0.35.

In FIG. 4 panel C, the template length was fixed to 32 read lengths toshow recovery as a function of the number of templates, from 2 to 1,024under the same conditions. Because these are simulations and the groundtruth was known, the proportion of exact matches are true and false canbe determined, and these numbers shown. All exact matches are connectedby an exact path, and for the conditions explored here, virtually alltrue exact matches are connected by a true path. At the maximum level ofcoverage, nearly all the true paths were determined, even for spans of2¹⁰ read lengths. For haplotype phasing, it is sufficient to have asingle true path, and this is attainable with high probability at alower coverage, 10× per template.

Methods—Partial Bisulfite Mutagenesis

Partial bisulfite mutagenesis was obtained in a single stranded phi x174 genomic DNA using the MethylEase™ Xceed Rapid DNA BisulphiteModification Kit (Human Genetic Signatures). The full conversionprotocol was modified by changing the incubation temperature to 73degrees Celsius (from 80 degrees Celsius) and the incubation time to 10minutes (instead of 45 minutes). Four regions were amplified to measurethe conversion rate.

Using the MethylEase™ Xceed Rapid DNA Bisulphite Modification Kit (HumanGenetic Signatures), the default incubation temperature (default 80degrees Celsius) and time (default 45 minutes) were modified. A range oftemperatures were tested (65, 73 and 80 degrees Celsius) and a range oftimes were tested (0, 3, 10, 30, 45 minutes). The remainder of theprotocol remains the same.

After bisulfite treatment, four specific regions of phi x were amplifiedwith “wobble” primers which contain a mixture of oligonucleotides suchthat the mutable positions are guanine (G) or adenine (A) for theforward primers and cytosine (C) or thymine (T) on the reverse primers.The addition of A-tails and ligatation of Illumina sequencing adaptersand sequence resulted in a sequence which was mapped back to the phi xgenome using a simple dictionary mapping method using a fully convertedgenome and fully converting all reads before mapping.

The best results (60% conversion) were obtained at an incubation of 10minutes at 73 degrees. A conversion rate of 20% was also obtained byincubating for 3 minutes at 73 degrees.

Methods—Custom Sequencing Primers

SEQ ID NO: 1: ACACTCTTTCCCTACACGACGCTCTTCCGATC*T

The sequencing primer (Seq ID p5.mC), wherein the cytosines (C) aremethylated, and wherein *T is a phosphorothioated thymine to protect theends from degradation by exonuclease.

SEQ ID NO.: 2: /5Phos/GATCGGAAGAGCGGTTCAGCAGGAATGCCGA*G

The sequencing primer (SEQ ID NO:2), wherein the cytosines (C) aremethylated, wherein *G is a phosphorothioated guanine to protect theends from degradation by exonuclease, and wherein 5Phos is a5′-phosphorylated, deoxyuridine-containing anchor-primer.

Example 4. Partial Bisulfite Mutagenesis

The experimental results from partial bisulfite mutagenesis of a phixtemplate genome (FIG. 6). The full conversion protocol was modified bychanging the incubation temperature to 73 degree Celsius (from 80degrees Celsius) and the incubation time to 10 minutes (instead of 45minutes). Four regions were amplified to measure the conversion rate.

FIG. 6 panel A depicts the rate at which each base is observed over allthe data for those positions with a coverage of at least 30 reads.Nearly all the C positions are at 40% C and 60% T. For each of the fourregions amplified, conversion patterns were compared between reads. Notall regions are equally well covered in the data, with 40-4500 reads.FIG. 6 panel B depicts the cumulative distribution of the conversionrate per read. FIG. 6 panel C depicts correlation in flips for allcytosines in the targeted region. It was determined that there none.FIG. 6 panel D depicts the data of FIG. 6 panel C as a histogram.

For the best amplified position, it is clear that partial conversion wasdetermined with a 60% flip rate randomly distributed throughout eachread.

Discussion—Examples 1-4

Presently, inferring the long-range structure of the DNA templates islimited by short read length. Accurate template counts suffer fromdistortions occurring during polymerase chain reaction (PCR)amplification. The utility of introducing random mutations in identicalor nearly identical templates was explored to create distinguishablepatterns that are inherited during subsequent copying. Simulations ofthe applications of this process were performed under assumptions oferror free sequencing and perfect mapping, employing cytosinedeamination as a model for mutation. The simulations demonstrate thatwithin readily achievable conditions of nucleotide conversion andsequence coverage, accurately counting the number of otherwise identicalmolecules as well as connecting variants separated by long spans ofidentical sequence can be achieved. Many potential applications include,transcript profiling, isoform assembly, haplotype phasing, RNAexpression analysis, copy number determination and de novo genomeassembly.

Counting varietal tags can be used to mitigate the effects ofamplification bias. While the original message is completelyrecoverable, the tag is confined to one end of the molecule such thatidentity and count can only be distinguished within one read length ofthe ends. Only reads that include the tag are useful in determiningcount and varietal tags provide no solution for assembly and assortment.

There are also protocols designed to aid sequence assembly of regionsthat are resistant due to base composition or repeat structure, whichrely on misincorporation of artificial nucleotides during amplification(Keith et al., 2004a; Keith et al., 2004b; Mitchelson, 2011).Implementation of this method requires many steps, and is not amenableto high throughput process, and in addition to such technical hurdles,is not suitable for counting as it loses track of template count.

Described herein are different approaches for counting and assemblingtemplates using template mutagenesis. The non-limiting examples hereindemonstrate by simulation that template mutagenesis can solve both theproblems of counting and assembly. For any application, the order ofoperation is mutagenesis first, followed by short- or long-range PCR,then fragmentation, if needed, and preparation of sequence libraries.Two classes of applications were explored. The first is countingspecific DNA or RNA molecules, for assessing genome copy number orprofiling a transcriptome (FIG. 1). The second is sequence assembly—forexample establishing haplotypes or distinguishing transcript isoforms(FIG. 2).

The simulations model partial bisulfite mutagenesis of single strandedDNA (Shortle and Botstein, 1983): each mutable position (or “bit”)converts (or “flips”) independently from a wild-type state to an alteredstate with a fixed probability (or “flip rate”). Performance wassimulated under a variety of reasonable parameters for read length andmutation rate, and over a range of template lengths and counts. Theresults are presented under an assumption of complete coverage to obtaina theoretical upper limit of performance and then consider theconsequences of sampling to various levels of coverage. In thesimulations, mutable positions are distributed uniformly throughout thetemplate such that each read contains the same number of bits (or “bitlength”). Sequence or mapping error are not presently incorporated.Variations to these assumptions and procedures are addressed herein.

A feasibility study and guide is presented for template mutagenesis asan enhancement to sequence based analysis. Such methods introduce randommutations to create distinguishable patterns in previously identical ornearly identical templates. These patterns are inherited in copies ofthe template, and portions of patterns remain in fragmented copies.Provided these fragments overlap and there is sufficient diversity inmutational patterns over a read length, the structure of each originaltemplate can be inferred, thus overcoming the loss of connectivityresulting from short read lengths. The accuracy of template countingalso improves, undistorted by biased amplification. The applications ofthis process were simulated under assumptions of error-free sequencing,perfect mapping, uniformly distributed reads, and uniform bitdistribution. The simulations suggest that within readily achievableconditions accurately counting the number of otherwise identicalmolecules can be determined, as well as connecting variants separated bylong spans of identical sequence. There are many potential applications,ranging from transcription profiling and isoform assembly to haplotypephasing and de novo genome assembly.

To illustrate one application of the results, characterizing single cellgene expression was considered. A mammalian cell has ˜350,000 mRNAtranscripts with an average length of 1,500 nucleotides (Alberts, 1994).Single strand cDNA would be randomly mutagenized using cytidinedeamination before amplification, fragmenting, and sequencing. Assuminguniformity of sequence coverage and uniformity of PCR amplification, 9million 100 base pair paired-end reads yields an average of 4× coverageper template. Given the typical distribution of cytidine in mammaliangenomes, 30 is a reasonable estimate for the number of mutable positionsin a read pair. From FIG. 3 panel C, it is clear that RNA templates canbe counted with near perfect accuracy for mRNA species of intermediateto scarce expression (<1,000 copies per cell).

Furthermore, many mammalian genes are alternatively spliced, resultingin a diversity of isoforms observable as different patterns of exoninclusion in the mRNA. At a read depth of 10× per template, or 23million reads, count can be determined not only transcripts per gene,but also expression at the level of individual isoforms: Even withoutthe additional information gained by observing alternative splicejunctions, a true path can be assembled from one end of the molecule tothe other. This can be accomplished for all but the most abundant genesand very long transcripts (greater than 6,000 nucleotides; FIG. 4, panelC). In contrast, varietal tagging can achieve accurate counting of genetranscripts, even long and abundant ones, but it is limited to labelingthe end of a molecule and so does not allow counting of isoforms orobserving sequence variants, except near the ends of transcripts. Thetwo methods, varietal tagging and mutagenesis can be seamlesslyintegrated, achieving the benefits of both methods.

Another direct application of template mutation is discriminatinghaplotypes in an individual. Using existing short-read technology, manyof the heterozygous positions can be readily identified; however,because the polymorphisms are typically more than one read length apart,the proper phasing of alleles cannot be determined. The followingprocedure for phase recovery is proposed: first perform partialdeamination on high-molecular-weight DNA with a 0.35 flip rate, thendilute it to 30 initial templates per region for each of the twostrands. Then amplifying with randomly primed PCR, and fragmenting asneeded for preparing sequencing libraries. Under the assumption ofuniformly distributed coverage (FIG. 4 panel B), it is observed thatcoverage of 10× per initial template would be sufficient to recoverphase information for variants separated by hundreds of kilobases.

Further, the ability to establish phase by this method depends on strongconcordance between the haplotypes and the reference genome. For thoseregions where the reference genome is a poor match, due to repeatcontent, large-scale rearrangements or novel sequence, the mutationpattern assembly algorithm will fail to generate consistent end-to-endassemblages. Although this presents a problem for direct inference byreference-matched phasing, it provides an opportunity for de novohaplotype assembly. The SUTTA algorithm (Narzisi and Mishra, 2011)assembles haplotypes from short-read data by scoring proposed localassemblies based on orthogonal data sources, such as coverage, matepairs, or physical maps. Template mutagenesis can help. Each localreference genome that SUTTA considers can also be assigned a score basedon the number of successful end-to-end mutation pattern assemblies overthe region. The result would be a de novo assembly over the human genomefor those difficult regions.

In the simulations, focus is on a specific form of mutation: theconversion of cytidine to uridine by deamination. This conversion can beachieved either chemically through bisulfite treatment (Shortle andBotstein, 1983) or enzymatically through activation-induced deaminase(Bransteitter et al, 2003). The advantages of deamination are thatconversion patterns are predictable. Moreover, because bisulfitetreatment is widely used in DNA methylation assays, the computationaltools for mapping deaminated sequence reads are readily available.Still, other methods of mutagenesis, such as depurination,transposition, alkylating agents, or inducing replication error in thefirst template copy might be useful in some contexts.

In the simulations perfect mapping was assumed. In practice, however,the ability to map reads might be somewhat degraded by templatemutation. For a deamination protocol, a standard practice is to map to areduced alphabet where all cytosines (“C”s) are converted to thymines(“T”s) in both the read and the reference, with two distinct referencesgenomes for each DNA strand (Krueger et al., 2011; Otto et al., 2012).Clearly, restricting to a smaller alphabet and doubling the referencegenomes impacts the ability to unambiguously map reads, however, theeffect is surprisingly mild (Krueger et al., 2012). If increased mappingefficiency is needed, the mapping algorithm can be augmented with aprobabilistic model of the flip rate to prioritize the most likelyalignments.

In the simulations no sequence error was assumed, but methods will benecessary for handling these. Aside from its effects on mapping,sequence error may reduce the ability to recover mutation patterns inthose cases where the error appears to flip a bit or reverse a flippedbit. Fortunately, sequence error is typically rare. Within a reasonablerange for flip rate, window size, and template count, sparse mutationalpatterns are expected, well separated so that no two patterns are verymuch alike. Sequence error will produce a pattern “nearby” anestablished pattern, and less well covered, and this signature can beused to discount those reads.

The simulations demonstrate that most applications work best for a lowinitial template count, less than a few thousand. This is not a problemfor many genomic applications and is close to ideal for single-cell RNAanalysis. If analysis of greater numbers of template molecules isdesired, for example during analysis of bulk mRNA, then aftermutagenesis of the first strand cDNA, multiple separate amplificationsreactions can be performed, each with low template count. The productsof each reaction can be tagged with barcodes, pooled and sequenced.

Example 5. Assembling Templates

A mutational protocol, muSeq, was established for partial bisulfiteconversion. The description of the method is very similar to thatestablished for the phiX samples discussed above. To establish theoperating characteristics of the protocol, a study was performed on aPstI digest of a human genome.

The experiment is described here. Genomic DNA is digested with arestriction enzyme (PstI). Fragments are end-repaired, adenylated, andthen ligated to bisulfite resistant sequencing adapters. These adaptersmatch the standard Illumina adapters, save that the cytosines arereplaced by methyl-cytosines. The sample is then treated with a standardkit for bisulfite treatment (MethylEasy Xceed Rapid DNA BisulphiteModification Kit Mix; Human Genetic Signatures.) Instead of incubatingthe sample for the standard of 80° C. for 45 minutes, 3, 6, and 9minutes at 73° C. were tested. One library using the standard 80° C. and45 minutes was also generated. The samples were sub-sampled, amplifiedand sequenced.

The resulting reads were mapped to the genome using an informaticspipeline designed for bisulfite sequence data. First the read-pairs arefully converted. For read 1, every C is converted to a T and for read 2,every G to an A. The converted read-pairs are then mapped twice, once toa genome where every C is converted to a T and once to a genome whereevery G is converted to A. The best mapping was assigned to the originalread-pair and the mapped genome recorded. Reads that map to theAGT-genome are called “original top” or “OT” and are templates derivedfrom the top strand of the initial restriction fragment. Similarly, thereads that map to the ACT genome are called “original bottom” or “OB.”Focus was on a 135 thousand fragments with high quality alignments inthe range of 150 to 400 base pairs.

The first observation, demonstrated in FIG. 7, is that the samemutational rate can be reliably established by setting the incubationtime. Also the mutational rates fall within the desired range.

Each restriction fragment/strand provides an opportunity to observemultiple reads derived from the same initial template. In error-freedata, one need only cluster reads that have precisely the same pattern.However, because of errors in sequencing and PCR amplification, a robustmethod was developed for joining reads derived from the same initialtemplate. Information was extracted from all convertible positions andthen cluster reads using a multi-scale clustering algorithm that workson pair-wise hamming distances [arXiv:1506.03072 (clustering methoddevised is available at arvix.org/abs/1506.03072)]. An example ofclustered reads at a single restriction fragment for the original topstrand is shown in FIG. 8.

To more carefully study the properties of the mutational sequencing, wesequenced the 6 minute conversion sample at great depth, on two lanes ofa nextSeq. When cytosine (or bit) positions were examined, the mutationswere found to be uncorrelated. Selecting 100,000 random pairs of bits,we compute the Fisher exact test and compare to the theoreticalexpectation that the two sites are independent. The resulting Q-Q plotis linear suggesting that the observed distribution does not divergefrom the null expectation and that deamination events are random andindependent (FIG. 9).

The conditions of digestion, amplification, and sequencing imply thatthe mean that coverage is not uniform for each locus. However, on thewhole, the counts of template from the autosome and the X chromosomediffer at a ratio of two to one, as expected in our male sample (FIG.10).

To study copy number in the context of identical sequence and coveragebias, we examine positions in the sample that are heterozygous.Heterozygous loci are nearly identical for sequence and so suffer thesame distortions. When the observed distribution was computed usingtemplate count, alleles were found to conform to the expected binomialdistribution (FIG. 11). The fit is far less good when the reads wereused instead of the template counts (FIG. 12). This suggests thattemplate count is superior for comparing copy number against a knowndifference. For example, counting the relative proportion of a rare SNPor determining copy number in the presence of a reference.

Finally, collapsing reads with the same pattern were found to reduce theerror in each template. By examining reads from homozygous positions,99.92% of consensus read positions were found to have greater than 80%of base pairs showing the homozygous base. This reduces typicallysequencing error rates by 100-fold.

Conversion of cDNA

The mutational protocol can be applied to cDNA as well. While this datais less well-studied, the preliminary results are very promising. Takingwhole RNA derived from cell lines, the mRNA was reverse transcribed withpoly-T and template switch oligo primers that are resistant to bisulfitemutation (methyl-cytosines substituted for cytosines). The resultingfirst strand cDNA were mutagenized with the muSeq protocol for 6 minutesat 73 C. The mutated strands were then sub-sampled, amplified,sonicated, repaired, and ligated to sequencing adapters, amplified andsequenced.

The resulting reads were then converted two ways (read 1 C→T, read 2G→A; and read 1 G→A, read 2 C→T) mapped to two versions of the humangenome using the STAR mapper (Dobin, et al STAR: ultrafast universalRNA-seq aligner, Bioinformatics. 2012) much as described above. The bestof the four maps were selected to assign to the original read.

Plots showing stacks of reads in the IGV viewer are shown in FIGS. 13and 14.

Preliminary results were obtained suggesting that long transcriptstructures can be recovered without mapping the reads at all but relyinginstead on methods of de novo assembly.

Example 6. Error Reduction

Using the high-coverage data with the 6 minute incubation and 5 roundsof post-bisulfite linear amplification, the properties of the consensussequences derived by clustering reads with the same mutation patternwere examined. Only genomic positions that are not bits and areconfidently homozygous in our sample were considered. For each consensussequence base, the proportion of reads reporting the homozygous base(homozygous base fraction) was determined. Greater than 99.99% ofconsensus bases have a majority of reads in agreement with thehomozygous base. Under more stringent requirements, it was found that99.92% of consensus bases match the homozygous base for greater than 80%of reads in the template (FIG. 13). These values bound the error rate assome of the disagreement is likely due to true somatic mutation.Nevertheless, these error rates are orders of magnitude below sequencererror, which is commonly observed at about one base error per positionper read.

Methodological Improvements

-   -   1. Direct mutagenesis of RNA. The RNA sequence can be directly        mutagenized before reverse transcription.    -   2. Mutagenesis on beads or surfaces. By binding template        molecules to a bead or surface, for instance by attaching        biotinylated primers to the templates and binding with        streptavadin beads, yield of post mutation templates may be        improved. Also multiple independent amplifications from the        bound targets to circumvent PCR error may be performed.    -   3. Binding to linear products of mutagenized templates to beads.        Similar to 2.

Example 7. Detecting One or More Variants Example 7A—Detecting TumorCells

A sample is obtained from a subject afflicted with cancer. The sample issubjected to a chemical mutagenesis as described herein. The mutagenizedsample is sequenced, aligned, mapped, and counted as described herein.

The presence of tumor cells in the sample is determined. Also,quantification of tumor cells in the sample is determined. Also, one ormore rare mutations in the sample is determined. Also, one or moresingle nucleotide polymorphisms in the sample is determined. Also,benign and malignantly transformed cells is distinguished.

Example 7B—Detecting a Small Load of Cancer DNA in the Presence of anExcess of Normal DNA

A sample is obtained from a subject afflicted with cancer. The sample issubjected to a chemical mutagenesis as described herein. The sample isfurther subjected to beads and/or varietal tags. The mutagenized sampleis sequenced, aligned, mapped, and counted as described herein.

The presence of tumor cells in the sample is determined. Also,quantification of tumor cells in the sample is determined. Also, one ormore rare mutations in the sample is determined. Also, one or moresingle nucleotide polymorphisms in the sample is determined. Also,benign and malignantly transformed cells is distinguished.

Example 7C—Detecting Prenatal Abnormalities

A sample is obtained from a pregnant female. The sample is subjected toa chemical mutagenesis as described herein. The mutagenized sample issequenced, aligned, mapped, and counted as described herein.

One or more rare mutations in a fetus is determined. Also, one or moresingle nucleotide polymorphisms in a fetus is determined. Also, one ormore chromosomal abnormalities in a fetus is determined.

Example 8. Sensitive Detection of Mutations

The error reduction described above in Example 6 is used in conjunctionwith the beads and/or varietal tags to obtain sequence counts for rarevariants.

Example 9. De Novo Transcriptome Assembly

A group of RNA transcipts in one cell or a population of cells isobtained. The group of RNA transcripts is subjected to a chemicalmutagenesis and sequenced as described herein.

An assembly algorithm is applied to the sequences of the group of RNAtranscripts. In some embodiments, the assembly algorithm may beSOAPdenovo-Trans, Velvet/Oases, Trans-ABySS, or Trinity transcriptomeassemblers. A transcriptome is obtained without mapping to a referencegenome.

Example 10. Metagenomics

A group of NAMs is obtained from genomes from a large variety oforganisms, wherein some of the organisms may be highly related. Thegroup of NAMs is subjected to a chemical mutagenesis and fragmentationas described herein.

The group of mutagenized fragments of NAMs is sequenced. In someembodiments, the sequencing may be metagenome sequencing, shotgunsequencing, or high-throughput sequencing.

An assembly algorithm is applied to the sequences of the group ofmutagenized fragments of NAMs. In some embodiments, the assemblyalgorithm may be Phrap, Celera, or Velvet/Oases assemblers. Independentgenomes are assembled.

Example 11. Python Programs Developed for a Wide Range of Simulations

fixed_window_finite_coverage

′′′

Performs simulations testing the recovery of template count over a fixedwindow for a range of flip_rates, read_lengths, and template_counts

assuming finite coverage

′′′

import sys

import numpy as np

from support_functions import getRandomPatterns, toBinary, tabprint

## fixes a random seed

seed=hash(“This is not a random seed.”)

np.random.seed(seed)

## input parameters

numsim=100 ## template simulations

covsim=100 ## sampling reads

read_lengths=[10, 20, 30]

flip_rates=np.linspace(0, 0.35, 8)[1:]

template counts=2**np.arange(1,11)

coverages=np.arange(1,17)

headings=[′R′, ‘p’, ‘T’, ‘coverage’, ‘mean distinct’]

print “\t”.join(headings)

## for each read length, flip rate and number of templates

for read_length in read lengths:

for flip_rate in flip rates:

-   -   for template_count in template counts:        -   ## perform numsim sims        -   unique_counts=[list( ) for_in range(len(coverages))]            ## for storing results    -   for sim in range(numsim):        -   ## generate template patterns and convert to binary            representation as int        -   templates=getRandomPatterns(template_count, read_length,            flip_rate)        -   templates_binary=np.array([toBinary(x) for x in templates])        -   ## for each level of coverage        -   for ind, coverage in enumerate(coverages):            -   num_reads=coverage*template_count            -   ## csim times            -   for csim in range(numsim):                -   ## sample uniformly from the templates                -   read_template=np.random.randint(0, template_count,                    num_reads)                -   ## consider just the observed templates                -   templates_observed=np.unique(read_template)                -   ## and ask how many are unique, saving the result                -   unique_count=len(np.unique(templates                    binary[templates observed]))                -   unique_counts[ind].append(unique_count)        -   ## after the sims are done        -   for ind, coverage in enumerate(coverages):        -   ## compute mean for each coverage level and print            -   mean_unique=np.mean(unique_counts[ind])            -   info=[read_length, flip_rate, template_count, coverage,                mean_unique]            -   print tabprint(info)            -   sys.stdout.flush( )                fixed window finite coverage                ′′′                Performs simulations testing the recovery of template                count over a fixed window for a range of flip rates,                read_lengths, and template_counts                assuming exhaustive coverage                ′′′                import sys                import numpy as np                from support_functions import getRandomPatterns,                toBinary, tabprint                ## fixes a random seed                seed=hash(“This is not a random seed.”)                np.random.seed(seed)                ## input parameters                numsim=100                read_lengths=[10, 20, 30]                flip_rates=np.linspace(0, 0.5, 11)[1:]                template_counts=2**np.arange(1,11)                headings=[‘p’, ‘T’, ‘mean_distinct’]                print “\t”.join(headings)                all_data=[ ]                all_info=[ ]                ## for each read length, flip rate and number of                templates                for read_length in read_lengths:

for flip_rate in flip_rates:

-   -   for template_count in template_counts:        -   unique_counts=[ ]        -   ## perform num sim simulations        -   for sim in range(numsim):            -   ## generate template patterns and convert to binary                representation as int            -   templates=getRandomPatterns(template_count, read_length,                flip_rate)            -   templates binary=np.array([toBinary(x) for x in                templates])            -   ## count how many are unique and save answer            -   unique_count=len(np.unique(templates_binary))            -   unique_counts.append(unique_count)        -   ## compute mean of sims and output results        -   mean_unique=np.mean(unique_counts)        -   info=[read_length, flip_rate, template_count, mean_unique]        -   all_info.append((read_length, int(100*flip_rate),            template_count))        -   all_data.append(unique_counts)        -   print tabprint(info)        -   sys.stdout.flush( )            many_windows_finite_coverage_assembly            ′′′            Performs simulations testing the recovery of template            assembly over a template of many read lengths for a fixed            flip_rate,            and a range of read_lengths, and template_counts,            and the number read lengths in the template            and a range of template coverage.            Scores edges based on likelihood of accidental agreement            and starting with the most unlikely edges,            adds a new edge if neither vertex already has a partner            in that direction with a better score.            Allows multiple edges when scores exactly agree            and so acts like the infinite case in the limit of total            coverage.            ′′′            import numpy as np            import sys            import time            from support_functions import getRandomPatterns, getMatch, \    -   getWordSpace, getOneCount, getScoreTable, \    -   greedyAssembly, pathScore, tabprint        from collections import defaultdict        ## input parameters        read_length=30        seed=hash(“This is not a random seed.”)        flip_rate=0.35        span_lengths=(2**np.arange(5,6))*read length        template_counts=2**np.arange(1,11)        coverages=2*np.arange(1,9)        ## fixes a random seed        np.random.seed(seed)        headings=[“template length”, “span length”, “read length”,    -   “flip rate”, “template count”, “coverage”,    -   “true exact match”, “false exact match”, “true path”,        “false path”, “time”]        print “\t”.join(headings)        ## for each read length, flip rate and number of templates        for span_length in span_lengths:

for template_count in template_counts:

-   -   ## we generate a template with a read length on either side of        the span    -   template_length=2*read length+span_length ## length of template    -   left_edge=read_length ## left edge of span (first read position        to not contain left mark)    -   right_edge=span_length ## right edge of span (last read position        to not contain right mark)    -   ## generate template patterns    -   templates=getRandomPatterns(template_count, template length,        flip rate)    -   ## make unique matched markers at the ends of the span    -   Apos_marks=np.arange(template_count)    -   Bpos_marks=np.arange(template_count)    -   Apos=read_length−1    -   Bpos=read_length+span length    -   marks={Apos:Apos_marks, Bpos:Bpos_marks}    -   ## compute matches between templates over all possible windows    -   match=getMatch(templates, read_length, marks=marks)    -   ## find space of unique reads    -   word_space=getWordSpace(match)    -   ## get count of ones, for every template, position, and window        size    -   one_count=getOneCount(templates, read_length)    -   ## get score lookup table    -   score_table=getScoreTable(read_length, flip_rate)    -   ## maximum read start position contained in the template    -   max_start=template_length−read_length+1    -   for ind, coverage in enumerate(coverages):        -   t0=time.time( )        -   num_reads=int(coverage*template_count*(float(template_length)/float(read_length)))        -   ## generate reads        -   read_template=np.random.randint(0, template_count,            num_reads)        -   read_start=np.random.randint(0, max_start, numreads)        -   ## re-index reads from template space to word space        -   ## word space assigns ambiguous reads to their least            template        -   read_index=[word_space[x] for x in zip(read_template,            read_start)]        -   ## reverse lookup to track which (index, start) derives from            which templates        -   ## needed to check for true paths        -   read_tracker=defaultdict(set)        -   for rstart, rtemplate, rindex in zip(read_start, read            template, read index):            -   read_tracker[(rstart, rindex)].add(rtemplate)        -   ## keep only unique read_index, read_start pairs        -   readlocs=np.zeros(shape=(template_count, max_start),            dtype=bool)        -   readlocs[read_index, read_start]=True        -   read_start, read_index=np.where(readlocs.T)        -   ## make a greedy assembly        -   edge_in, edge_out, inscore=greedyAssembly(read_start,            read_index, match, one_count, score_table, read_length,            template_length)        -   ## score the result        -   true_exact_match, false_exact_match, true_path,            false_path=pathScore(edge_in, edge_out, read_start,            read_index, left_edge, right_edge, read_tracker)        -   ## and output        -   timediff=time.time( )−t0        -   info=[template length, span_length, read_length, flip_rate,            template_count, coverage, true_exact_match, false_exact            match, true_path, false_path, timediff]        -   print tabprint(info)        -   sys.stdout.flush( )            many_windows_finite_coverage_counting            ′′′

Performs simulations testing the recovery of template count over atemplate of many read lengths for a fixed flip_rate,

and a range of read_lengths, and template counts,

and the number read lengths in the template

and a range of template coverage.

For speed, finds minimum cover in overlapping intervals of length 2R andtakes the maximum over all such windows.

We can do this because the maximum antichain

must be contained in an interval of length R

and must therefore be included in at least one interval.

The provides a speed advantage for long templates.

′′′

import numpy as np

from support_functions import getRandomPatterns, getMatch, \

-   -   getWordSpace, overlapMatch, tabprint        from hoperoft_karp import bipartiteMatch        import bisect        import sys        ## input parameters        seed=hash(“This is not a seed.”)        flip_rate=0.35        read_lengths=[10,20,30]        template_factors=(2**np.arange(1,5))        template_counts=2**np.arange(1,11)        coverages=np.arange(1, 11)        ## fixes a random seed        np.random.seed(seed)        headings=[“template_length”, “read_length”, “flip_rate”,        “template_count”, “coverage”, “min_cover”]        print “\t”.join(headings)        ## for each read length        for read_length in read_lengths:

## for each template factor

for template_factor in template factors:

-   -   template_length=read_length*template_factor    -   ## for each number of templates    -   for template_count in template_counts:        -   ## generate template patterns        -   templates=getRandomPatterns(template_count, template_length,            flip_rate)        -   ## compute matches between templates over all possible            windows        -   match=getMatch(templates, read_length)        -   ## find space of unique reads        -   word_space=getWordSpace(match)        -   ## maximum read start position contained in the template        -   max start=template length−read_length+1        -   for ind, coverage in enumerate(coverages):            -   num_reads=int(coverage*template_count*(float(template_length)/float(read_length)))            -   ## generate reads            -   read_template=np.random.randint(0, template_count,                num_reads)            -   read_start=np.random.randint(0, max_start, num reads)            -   ## re-index reads from template space to word space            -   ## word space assigns ambiguous reads to their least                template            -   read_index=[word_space[x] for x in zip(read_template,                read_start)]            -   ## keep only unique read_index, read_start pairs                readlocs=np.zeros(shape=(template_count, max_start),                dtype=bool)            -   readlocs[read index, read start]=True            -   read_start, read index=np.where(readlocs.T)            -   ## we will look at overlapping intervals of length 2R            -   s=np.arange(0, template_length-read_length,                step=read_length)            -   e=s+2*read_length+1            -   ## appropriate indices in read_starts            -   A=[bisect.bisect_left(read_start, x) for x in s]            -   B=[bisect.bisect_left(read_start, x) for x in e]            -   ## set minimum cover to 0            -   min_cover=0            -   ## for each interval            -   for (a,b) in zip(A, B):                -   ## get read starts and indices                -   rs=read_start[a:b]                -   ri=read_index[a:b]                -   ## get graph of all overlapping matching edges                    overmatch_graph=overlapMatch(rs, ri, match,                    read_length, template_length)                -   ## and all edges that fail to overlap                -   for i, j in                    zip(*np.where(np.less_equal.outer(rs+read_length,                    rs))):                -    overmatch_graph[i].add(j)                -   ## compute bipartite matching                -   M, A, B=bipartiteMatch(overmatch_graph)                -   ## and take difference with vertex set for minimum                    cover                -   mc=len(rs)−len(M)                -   ## update mincover to reflect maximum value                -   min_cover=max(mc, min_cover)            -   ## and output the results            -   info=[template_length, read_length, flip_rate,                template_count, coverage, min_cover]            -   tabprint(info)            -   sys.stdout.flush( )                many_windows_infinite_coverage                ′′′                Performs simulations testing the recovery of template                count and assembly                over a template of many read lengths                for a range of flip_rates, read_lengths, and                template_counts assuming exhaustive coverage                ′′′                #!/data/software/local/bin/python                import sys                import numpy as np                from support_functions import getRandomPatterns,                templatesToWindows,                \    -   generateCompleteDAG_trimmed, tabprint from hoperoft_karp import        bipartiteMatch        ## input parameters        read_lengths=[10, 20, 30]        template_factors=(2**np.arange(1,5))        flip_rates=np.linspace(0, 0.35, 8)[1:]        template_counts=2**np.arange(1,11)        seed=hash(“This is not a random seed.”)        ## fixes a random seed        np.random.seed(seed)        headings=[′L′, ‘R’, ‘p’, ‘T’, ‘perfect_paths’, ‘min_cover’]        print “\t”.join(headings)        for read_length in read_lengths:

## for each read length, flip rate and number of templates

for template_factor in template_factors:

-   -   template_length=read length*template_factor    -   for flip_rate in flip_rates:        -   for template_count in template_counts:            -   ## generate template patterns            -   templates=getRandomPatterns(template_count,                template_length, flip_rate)            -   ## generate reads as binary representations            -   read=np.array(templatesToWindows(templates,                read_length))            -   ## generate overlaps (read_length−1 windows)            -   overlap=np.array(templatesToWindows(templates,                read_length−1))            -   ## counts the number of overlap collisions for each                template at each window            -   overlap_count=np.array([np.sum(np.equal.outer(x, x), 0)                for x in overlap.T]).T            -   ## number of overlap collisions in each template            -   collapse_count=np.sum(overlap_count>1, axis=1)            -   ## the number of unambiguous paths            -   ## (those with no overlap collisions at any position)            -   perfect_paths=np.sum(collapse_count==0)            -   ## generate bipartite graph from reads and overlaps                (trimmed of simple nodes)            -   graph=generateCompleteDAG_trimmed(read, overlap)            -   ## and determine maximal bipartite match to get minimum                vertex cover            -   M, A, B=bipartiteMatch(graph)            -   ## the size of the minimum cover            -   ## equals the number of items lacking partners in the                match            -   min_cover=len(graph)−len(M)            -   ## and report            -   info=[template_length, read_length, flip_rate,                template_count, perfect_paths, min_cover]            -   print tabprint(info)            -   sys.stdout.flush( )                support_functions                ′′′                A collection of function common to many of the                simulation tests. Includes methods for generating random                template patterns                and random reads from template patterns                as well as some of the directed graph functions                ′′′                import numpy as np                import heapq                import bisect                from collections import defaultdict, Counter                def tabprint(A):

′′′returns all elements of A, converted into strings and joined bytabs.′′′

return “\t”.join(map(str, A))

def toBinary(X):

′′′converts a bool array to its binary integer (or long)representation′′′

return int((len(X)*‘% d’) % tuple(X[::−1]),2)

def bitCount(int_type):

′′′returns the number of bits that are on in the word′′′

return bin(int_type).count(“1”)

def toBitCount(inputArray):

′′′

takes an array of binary ints and

returns an array of the same size containing the bitcount

′′′

ans=np.zeros_like(inputArray)

for index, row in enumerate(inputArray):

-   -   ans[index]=[bitCount(x) for x in row]

return ans

def getRandomPatterns(number_of_templates, length_of_template,probability_of_flip):

′′′generate random bit patterns′′′

return np.random.random(size=(number_of_templates,length_of_template))<probability_of_flip

def templateToWindows(template, window_size):

′′′convert a template pattern into its sequence of window_sizesubwords′′′

max_start=len(template)−window size+1

return [toBinary(template[i:(i+window size)]) for i in range(max_start)]

def templatesToWindows(templates, window_size):

′′′convert a set of template patterns into their sequence of window_sizesubwords′′′

return [templateToWindows(template, window_size) for template intemplates]

def getMatch(templates, R, marks=None):

′′′

function to determine exact agreement between all pairs of templates

starting at a position and extending up to some length.

in particular:

shape: template, template, position, window_size

and match[T1, T2, P, W] returns the truth value for the statement

T1 and T2 match for every position on the window P:(P+W)

also set to accept a set of special marks.

marks is None or a dictionary

key is a position and marks[pos] are the mark values for each templatein order.

for example, if each of four templates has a unique mark at pos 10:

marks[10]=[0,1,2,3]

if there are two biallelic loci Aa and Bb with true phase AB, ab, mighthave:

marks[10]=[A, a, a, A]

marks[70]=[B, b, b, B]

′′′

T, L=templates.shape

## does a pair t1, t2 mismatch at a position p?

mismatch_pos=np.zeros(shape=(T,T,L), dtype=‘bool’)

for i in range(T):

-   -   mismatch_pos[i]=np.logical xor(templates[i], templates)

## if there are special marks:

if marks !=None:

-   -   ## then at each position:    -   for pos in marks:        -   markers=marks[pos]        -   ## put a mismatch wherever markers disagree        -   np.logical_or(mismatch_pos[:,:,pos],            np.not_equal.outer(markers, markers),            out=mismatch_pos[:,:,pos])

## does a pair t1, t2 mismatch on a window p:(p+w)?

mismatch=np.zeros(shape=(T,T,L, R+1), dtype=′bool′)

for k in range(0, R):

-   -   toEnd=L−k    -   np.logical_or(mismatch[:, :, :toEnd, k], mismatch_pos[:, :, k:],        out=mismatch[:, :, :toEnd, k+1])

return np.logical not(mismatch)

def getWordSpace(match):

′′′

At each position, converts from template index to word index.

if each word is unique at a position, then they are identical.otherwise, each word is assigned its lowest index template.

This is necessary for handling collisions between reads

for indexing graph vertices.

′′′

template_count=match.shape[0] ## number of templates

read_length=match.shape[3]−1 ## length of read

max_start=match.shape[2]−read_length+1 ## maximum read start position

word_space=np.zeros(shape=(template_count, max_start), dtype=int)

## for each possible start position

for pos in range(max_start):

-   -   ## for each template    -   for t in range(template_count):        -   ## set word space to the least index with a match        -   x=match[t, :, pos, −1]        -   word_space[t, pos]=np.argmax(x)

return word_space

def getOneCount(templates, read_length, dtype=‘int8’):

′′′

-   -   Number of bits flipped to ‘1’ for every template, every        position, and every window up to the read length.    -   Has shape: template, position, window_size and one_count[T, P,        W] returns the number of flipped positions in the window P:(P+W)        in template T.

′′′

T, L=templates.shape

one_count=np.zeros(shape=(T, L, read_length+1), dtype=dtype)

## computed iteratively

for k in range(0, read_length):

-   -   toEnd=L−k    -   one_count[:, :toEnd, k+1]=one_count[:, :toEnd, k]+templates[:,        k:]

return one_count

def getScoreTable(read_length, flip_rate):

′′′

Assigns a value to an edge based on the probability that it isaccidental.

These values depend on the number of ones, the length of the word,

and the probability of a one.

This function generates a lookup table that keeps the values for:

length of overlap, number of ones for a fixed flip rate.

′′′

log p=np.log(flip_rate)

log q=np.log(1−flip_rate)

M=np.arange(read_length+1)[:, np.newaxis]

K=np.arange(read_length+1)[np.newaxis, :]

score_table=K*log p+(M−K)*log q

return score_table

def generateCompleteDAG(read, overlap):

′′′

function generates the enhanced directed acyclic graph

assuming perfect data——all reads and all overlaps are known.

The primary DAG has vertices for each read

and an edge from A to B if pos(B)=pos(A)+1

and the reads agree on the overlap.

The complete DAG has the same vertices,

and an edge from A to B if there exists a path from A to B

in the primary DAG

The reason for enhanced DAG is this:

Applying bipartite matching to the primary DAG gives a minimumvertex-disjoint cover.

Applying bipartite matching to the enhanced DAG gives a minimum vertexcover.

′′′

graph=defaultdict(set)

max_start=read.shape[1]−2

## iterate backwards from the next to last position to the first

for position in range(max_start, −1, −1):

-   -   ## compute which overlaps agree    -   overlap_match=np.equal.outer(overlap[:, position+1], overlap[:,        position+1])    -   ## for all pairs that share an overlap    -   for x,y in zip(*np.where(overlap_match)):        -   source=(position, read[x, position]) ## define source        -   target=(position+1, read[y, position+1]) ## and target        -   ## to the source, add the target (and the target's targets)        -   graph[source].update(graph[target])        -   graph[source].add(target)

return graph

′′′

Similar to the complete DAG

but built from a primary DAG that is free of simple nodes.

Y is a simple node if

for X→Y→Z

there is only one such X and Z and X has no other out-edges.

Then any path through Y must be from X and to Z

and any path through X must continue through Z.

So Y can be removed without changing the vertex cover.

Repeat until there is no such Y in the graph.

′′′

def generateCompleteDAG_trimmed(read, overlap):

graph=defaultdict(set)

max_start=read.shape[1]−2

in_edge=defaultdict(set)

out_edge=defaultdict(set)

## iterate backwards from the next to last position to the first

for position in range(max_start, −1, −1):

-   -   ## compute which overlaps agree    -   overlap_match=np.equal.outer(overlap[:, position+1], overlap[:,        position+1])    -   ## for all pairs that share an overlap    -   for x,y in zip(*np.where(overlap_match)):        -   source=(position, read[x, position]) ## define source        -   target=(position+1, read[y, position+1]) ## and target        -   out_edge[source].add(target)        -   in_edge[target].add(source)

## remove simple edges

for x in in edge.keys( ):

-   -   if len(in_edge[x])==1 and len(out_edge[x])==1:        -   source, target=list(in_edge[x])[0], list (out_edge [x]) [0]        -   if len(out_edge[source])==1:            -   out_edge[source].remove(x)            -   in_edge[target].remove(x)            -   out_edge[source].add(target)            -   in_edge[target].add(source)            -   del in_edge[x]            -   del out_edge[x]:

for x in in edge.keys( )

-   -   if len(in_edge[x])==0:        -   del in_edge[x]

for x in out_edge.keys( ):

-   -   if len(out_edge[x])==0:        -   del out_edge[x]

node_list=sorted(out_edge.keys( )[::−1]

## build graph

for x in node list:

-   -   graph[x].update(out_edge[x])    -   for y in out_edge[x]:        -   graph[x].update(graph[y])

return graph

def generatePrimaryDAG(read, overlap):

′′′

function generates the primary directed acyclic graph

assuming perfect data——all reads and all overlaps are known.

The primary DAG has vertices for each read

and an edge from A to B if pos(B)=pos(A)+1

and the reads agree on the overlap.

′′′

graph=defaultdict(set)

max_start=read.shape[1]−2

## iterate backwards from the next to last position to the first

for position in range(max_start, −1, −1):

-   -   ## compute which overlaps agree    -   overlap_match=np.equal.outer(overlap[:, position+1], overlap[:,        position+1])    -   ## for all pairs that share an overlap    -   for x,y in zip(*np.where(overlap match)):        -   source=(position, read[x, position]) ## define source        -   target=(position+1, read[y, position+1]) ## and target        -   graph[source].add(target)

return graph

def overlapMatch(read_start, read_index, match, read_length,template_length):

′′′

First, assumes that read_starts are sorted. Safe in this code.

Uses bisect to make an index for the read_start positions.

Then iterating over each position,

checks for all overlaps in blocks to take advantage of vectoroperations.

Returns a list of all read pairs that overlap and agree on that overlap.

′′′

max_start=template_length−read_length+1

## number of reads

num_nodes=len(read start)

## index into the start positions (assumed sorted)

index_to_start=np.array([bisect.bisect_left(read_start, x) for x innp.arange(template_length)]+[num_nodes])

overmatch_graph=defaultdict(set)

## for each read start position in the template

for a_pos in np.arange(max_start):

-   -   ## get the upper and lower index bounds for a_pos    -   low, high=index_to_start[a_pos:(a_pos+2)]    -   ## get the upper index bound for possible overlaps    -   end=index_to_start[a_pos+read_length]    -   ## a is everything at this position (a_pos)    -   ## b is everything from the next position on,    -   ## that would overlap reads at this position.    -   a_template=read_index[low:high]    -   b_template=read_index[high:end]    -   b_pos=read_start[high:end]    -   overlap_length=read_length−b_pos+a_pos    -   hasmatch=match[:,b_template,b_pos,overlap_length][a_template]    -   ## anything that has a match, goes into the list for i, j in        zip(*np.nonzero(hasmatch)):        -   overmatch_graph[low+i].add(high+j)

return overmatch_graph

def pathScore(edge_in, edge_out, read_start, read_index, left_edge,right_edge, read_tracker):

′′′

Determine the number of exactly matched tag pairs.

Of those, how many are tags from the same original template (“truematches”)?

Of the true matches, how many admit a “true path”

in which all the reads on the path could have derived from the sameinitial template?

True matched pairs that admit true paths

are well-assembled templates in which all template information,

such as SNP phasing or intron exclusions,

is recoverable from mutation pattern assembly alone.

A true match with not true path correctly pairs the ends,

but may have some errors in between.

The function expects that the read_starts are sorted,

so that out_edges always point to elements lower in the list.

′′′

num_nodes=len(edge_in)

in_count=np.array([len(x) for x in edge_in]) ## number in

out_count=np.array([len(x) for x in edge_out]) ## number out

path_start=np.where(in_count==0)[0] ## starts have no in

path_end=np.where(out_count==0)[0] ## ends have no out

## for each path start, add marks that are to the left of the edge

## i.e. contain the left marker

start_marks=[set( ) for_in range(num_nodes)]

for pstart in path_start:

-   -   if read_start[pstart]<left_edge:        -   start_marks[pstart].add(pstart)

## propagate through the graph

for source in range(num_nodes):

-   -   for target in edge_out[source]:        -   start_marks[target].update(start_marks[source])

## and out the end,

## track how many times a start node is seen an end node

## (beyond the right_edge, i.e. containing a right mark)

start_counter=Counter( )

for pend in path_end:

-   -   if read_start[pend]>right_edge:        -   for pstart in start_marks[pend]:            -   start_counter[pstart]+=1

def hasTruePath(tindex, node, pend):

-   -   ′′′    -   determines if there is a true path from node to pend,    -   i.e. a path that traverses nodes supported by reads that are in        the same template.    -   Some nodes may be referred by multiple reads from different        templates——    -   if the mutation patterns are degenerate for more than a read        length.    -   ′′′    -   if node==pend:        -   return True    -   elif tindex not in read_tracker[read_start[node],        read_index[node]]:        -   return False    -   else:        -   for next_node in edge out[node]:            -   if hasTruePath(tindex, next_node, pend):                -   return True        -   return False

## now count up true and false exact matches and true paths

true_exact_match=0

false_exact_match=0

true_path=0

false_path=0

## for each end

for pend in path_end:

-   -   ## if it is beyond the right edge (contains a right mark)    -   if read_start[pend]>right_edge:        -   ## does it have exactly one start?        -   if len(start_marks[pend])==1:            -   pstart=list(start marks[pend])[0]            -   ## does that start have just one end?            -   if start_counter[pstart]==1:                -   ## do they agree on template index?                -   if read_index[pstart]==read_index[pend]:                -    ## test for true path                -    tindex=read_index[pstart]                -    has_true_path=hasTruePath(tindex, pstart, pend)                -    if has true path:                -    true_path+=1                -    else:                -    false_path+=1                -    true_exact_match+=1                -   else:                -    false_exact_match+=1

return_true_exact_match, false_exact_match, true_path, false_path

def scoredOutEdgeHeaps(read_start, read_index, match, one_count,score_table, read_length, template_length):

′′′

Function returns a list of ordered heaps of out edges for each vertex.

Two reads have an edge if they share an overlap, agree on that overlap

and are not at the exact same position.

The score for an edge is the log likelihood of an accidental overlap

of that length and with that number of flipped bits.

Used by greedy assembly.

′′′

max_start=template_length−read_length+1

## number of reads

num_nodes=len(read_start)

## index into the start positions (assumed sorted)

index_to_start=np.array([bisect.bisect_left(read_start, x) for x innp.arange(template_length)]+[num_nodes])

## for each node, make a heap to store out edges

outedge_heaps=[[ ] for_in range(num_nodes)]

## for each read start position in the template

for a_pos in np.arange(max_start):

-   -   ## get the upper and lower index bounds for a_pos    -   low, high=index_to_start[a_pos:(a_pos+2)]    -   ## get the upper index bound for possible overlaps    -   end=index_to_start[a_pos+read_length]    -   ## a is everything at this position (a_pos)    -   ## b is everything from the next position on,    -   ## that would overlap reads at this position.    -   a_template=read_index[low:high]    -   b_template=read_index[high:end]    -   b_pos=read_start[high:end]    -   overlap_length=read_length−b_pos+a_pos    -   hasmatch=match[:,b_template,b_pos,overlap_length][a_template]    -   for i, j in zip(*np.nonzero(hasmatch)):        -   bits=overlap_length[j]            ## bits of overlap    -   flipped=one count[b_template[j], b_pos[j], bits]        ## how many are on    -   score=score_table[bits, flipped]        ## lookup score    -   heapq.heappush(outedge_heaps[low+i], (score, high+j))        ## push into heap

return outedge_heaps

def greedyAssembly(read_start, read_index, match, one_count,score_table, read_length, template_length):

′′′

Returns the edges of a greedy assembly.

Iterate through the edges in order of least likely read coincidence.

Join an edge A to B if neither A nor B already have an out (in) partner

with a better score.

′′′

num_nodes=len(read_start)

## getting edge out heaps

outedge_heaps=scoredOutEdgeHeaps(read_start, read_index, match,one_count, score_table, read_length, template_length)

## keep the best in edge score for each vertex

inscore=np.zeros(num_nodes, dtype=float)

## list of lists for storing edges

edge_in =[list( ) for_in range(num_nodes)]

edge_out=[list( ) for_in range(num_nodes)]

## build a heap for the vertices

vertex heap=[ ]

for i in range(num_nodes):

-   -   try:        -   ## ordered by the score of their top element        -   item=(outedge_heaps[i][0][0], i)        -   heapq.heappush(vertex_heap, item)    -   except:        -   ## should it exist        -   continue

## then start working through the vertex heap

while len(vertex heap)>0:

-   -   ## find the out vertex with the best score    -   score, out_vertex=heapq.heappop(vertex_heap)    -   outheap=outedge_heaps[out_vertex]    -   ## and then pop its heap to get all its partners with that best        score    -   foundPartner=False    -   while len(outedge_heaps[out_vertex])>0:        -   if outheap[0][0]==score: ##            if best element has score    -   in_score, in_vertex=heapq.heappop(outheap) ## pop it off the        heap    -   ## check if its a good score for the in vertex    -   if score <=inscore[in_vertex]:        -   foundPartner=True ##            found a partner    -   edge_in[in vertex].append(out_vertex) ##        add the edges    -   edge_out[out_vertex].append(in_vertex)    -   inscore[in_vertex]=score ##        and record the score    -   else:        -   break    -   ## if the vertex did not find a partner, it returns to the        vertex heap    -   ## with its best score.    -   if not foundPartner:        -   try:            -   item=(outedge heaps[out_vertex][0][0], out_vertex)            -   heapq.heappush(vertex_heap, item)        -   except:            -   continue

return edge_in, edge_out, inscore

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What is claimed:
 1. A method for determining the number of nucleic acidmolecules (NAMs) in a group of NAMs, comprising i) obtaining anamplified and mutagenized group of NAMs that was produced by a.subjecting the group of NAMs to a chemical mutagenesis which mutatesonly select nucleic acid bases in the group of NAMs at a rate of 10% to90% thus forming a group of mutagenized NAMs (mNAMs), and b. amplifyingthe group of mNAMs; ii) obtaining sequences of the mNAMs in the group ofamplified mNAMs; and iii) counting the number of different sequencesobtained in step (ii) to determine the number of unique mNAMs in thegroup of mNAMS, thereby determining the number of NAMs in the group ofNAMs.
 2. The method of claim 1, wherein obtaining sequences in step (ii)comprises obtaining composite sequences produced by assembling sequencereads of the mNAMs by a) aligning the sequence reads according tomatching mutation patterns in overlaps of the sequence reads, therebyobtaining composite sequences, and b) mapping the composite sequences;and wherein step (iii) comprises counting the number of jointlyoverlapping different composite sequences obtained in step (ii).
 3. Themethod of claim 1, wherein a sub-group of NAMs in the group of NAMs isdetermined to have substantially the same nucleotide sequence, whereinthe sub-group of NAMs is determined to have nucleotide sequences thatare at least 95, 96, 97, 98, 99, 99.1, 99.2, 99.3, 99.4, 99.5, 99.6,99.7, 99.9, or 99.9% identical; or wherein the nucleotide sequences of asub-group of NAMs comprise a stretch of consecutive nucleotides having asequence which includes at least two mutable positions and is i)identical to the sequence of a stretch of consecutive nucleotides withinanother NAM within the sub-group of NAMs, or ii) determined to have atleast 95, 96, 97, 98, 99, 99.1, 99.2, 99.3, 99.4, 99.5, 99.6, 99.7,99.9, or 99.9% identical to the sequence of a stretch of consecutivenucleotides within another NAM within the sub-group of NAMs.
 4. Themethod of claim 1, wherein the counting comprises counting the number ofdifferent sequences that are determined to have substantially the samesequence except for their mutable positions, thereby determining thenumber of NAMs in the group of NAMs that had substantially the samesequence; or wherein the counting comprises counting the number ofdifferent sequences which lack substantially the same sequence in anystretch including at least two mutable positions, thereby determiningthe number of NAMs without substantially the same sequence in the groupof NAMs.
 5. The method of claim 1, wherein a) the mutagenesis is bycytosine deamination, b) each mutable position of the NAMs comprises acytosine and the cytosine (C) is mutated to a uracil (U) or a thymine(T), c) each NAM in the group of NAMs has a unique primer at its 5′ endand another unique primer at its 3′ end, d) the NAM is within a mixtureof DNA or RNA extracted from a cell, wherein the DNA or RNA extractedfrom the cell has been fragmented, e) the NAM is a DNA molecule, or anRNA molecule, f) one or more NAMs in the group of NAMs has a length ofone sequencing read length, g) one or more NAMs in the group of NAMs hasa length of two or more sequencing read lengths, and/or h) the number ofNAMs in the group of NAMs is about 2, 3, 4, 5, 6, 7, 8, 9, 10, or10-10000, wherein if the number of NAMs in the group of NAMs is greaterthan 10000, then diluting the group of NAMs.
 6. The method of claim 1,further comprising the step of tagging each NAM or copy thereof, whereinthe tag lacks a nucleotide that is mutable by the mutagenesis.